Algebra and Trigonometry (1st Edition)

Chapter 9

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Consider the left hand side of the equation , and ... more

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Consider the left-hand side of the equation , and ... more

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Consider the left-hand side of the equation , and ... more

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; In the expression , 25 is the square of 5. ... more

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Consider the left-hand side of the equation , and ... more

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All the functions are even. ; For the function , ... more

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The graph of between the interval will form, ... more

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is the reciprocal of the cosine function . ... more

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Divide the identity by on both the sides. ... more

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; Simplify the expression using .So, ; more

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; Simplify the expression using the formulas ... more

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. ; In the expression , rewrite and in terms of ... more

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. ; Simplify the expression using , , and , and ... more

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; Simplify the numerator by applying following ... more

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3. ; Rewrite the expression by substituting as ... more

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-1. ; Express as the inverse of , and simplify ... more

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. ; First simplify the numerator by applying the ... more

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; Express the secant function as the inverse of ... more

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-1. ; Express the cosecant and secant functions as... more

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. ; The tangent function can be re-written as the ... more

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; Convert the tangent and cotangent function into... more

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. ; Convert the tangent and cotangent function ... more

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. ; Convert the tangent  function into sine and ... more

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. ; Convert the  cotangent function into sine and ... more

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; Convert the  sine function into cosecant ... more

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. ; Convert the secant, cosecant, tangent, and ... more

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. ; Convert the cosecant function into sine ... more

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; To simplify the expression , subtract both the ... more

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; To simplify to , convert the tangent function ... more

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; Convert the secant function into sine and ... more

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; Convert the secant function into cosine ... more

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; Convert the cotangent function into sine and ... more

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; Convert the cotangent function into sine and ... more

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The identity can be verified by transforming one ... more

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The identity can be verified by transforming one ... more

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The identity can be verified if each if the ... more

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The identity can be verified by transforming one ... more

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The identity can be verified by transforming one ... more

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The identity can be verified by transforming one ... more

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The identity can be verified by transforming one ... more

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. ; Simplify the left hand side by applying the ... more

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Hence, the identity has been disproved. ; Simplify... more

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Hence, the identity has been proved. ; Simplify ... more

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Hence, the identity has been proved. ; Simplify ... more

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The given identity is false and the correct ... more

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The given identity is true. ; Simplify the left-... more

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The given identity is false and the correct ... more

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; Use the cosine sum formula to expand . ; Now, ... more

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; Write as .Use the cosine sum formula to ... more

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; Use the sum formula for the tangent to expand... more

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; Since the cofunction of is , substitute to ... more

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Use the difference formula for tangent in the ... more

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Conjunction identities are applied in the case of ... more

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No, can be written as and , and both the cosines... more

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The difference formula for sine and cosine ... more

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; Write as .Use the sum formula for cosine to ... more

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; Write as .Use the difference formula for ... more

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. ; Write as .Use the sum formula for sine to ... more

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. ; Write as .Use the sum formula for sine to ... more

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. ; Write as .Use the sum formula for tangent to... more

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. ; To evaluate , split the angle such that the ... more

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. ; Use the sum formula of sine, that is to ... more

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. ; Use the difference formula of sine, i.e., , to... more

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. ; Use the difference formula of cosine, i.e... more

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. ; Use the sum formula of cosine, i.e., , to ... more

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. ; Use the cofunction identity and simplify the ... more

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. ; Use the cofunction identity and simplify the ... more

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. ; Use the cofunction identity and simplify the ... more

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. ; Use the cofunction identity and simplify the ... more

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. ; Compare the expression with the difference ... more

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. ; Compare the expression with the difference ... more

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. ; The sum formula for sine is . Here and , ... more

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. ; and , where , are the angles which lie in ... more

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. ; The expression is in the form of difference ... more

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. ; The expression, is of the form of sum formula... more

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. ; The expression, is of the form of difference ... more

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. ; Use the cofunction identity and simplify the ... more

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. ; Use the difference formula of sine which is ... more

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. ; Use the sum formula of tangent, which is , and... more

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.  ; Use the sum formula of sine, which is and ... more

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; To simplify the expression , use .  Graph the ... more

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; Use to simplify the expression .  Graph the ... more

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  ; To simplify the expression , use .  Graph the ... more

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; To simplify the expression , use .  Graph the ... more

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Both the functions and are the same. ; Use the ... more

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The functions and are different. ; Use the sum ... more

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The functions and are different. Try . ; Use the... more

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Both the functions and are the same. ; Use the ... more

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Both the functions and are the same. ; Use the ... more

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Both the functions and are the same. ; Graph the... more

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The functions and are different. Try . ; Graph ... more

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Both the functions and are the same. ; Use the ... more

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The functions and are different. Try . ; Graph ... more

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or 0.9659 ; To find the value of , use the ... more

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or -0.2588 ; To find the value of , use the ... more

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or -0.9659 ; To find the value of , use the ... more

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or 0.9659 ; To find the value of , use the ... more

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or -0.2679 ; To find the value of , use the ... more

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Step 1: Use the sum formula for tangent to expand ... more

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Step 1: Use the sum formula for tangent to expand ... more

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Simplify the left-hand side expression using the ... more

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Consider the left-hand side expression and ... more

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Consider the left-hand side expression , and ... more

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The left-hand side expression can be rewritten is... more

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The left-hand side expression can be expressed as... more

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Simplify the left-hand expression using the ... more

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Consider the left-hand side expression , and ... more

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The summation of , , and is . Therefore, .... more

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. ; First, calculate the value of using the ... more

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Consider the left-hand side of the expression , ... more

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Consider the double angle formula of the cosine ... more

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Consider the left-hand side of the expression and... more

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; Calculate the value of using the trigonometric... more

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Replace the variable with in the formula to ... more

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The function can be expressed as .Then, simplify ... more

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The two formulas for from the equation can be ... more

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In the given expression , when the denominator is ... more

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; Use the half angle formula to calculate .Since... more

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; Use the half angle formula to calculate .Since... more

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; Use the formula to simplify the mathematical ... more

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; Use the formula to simplify the mathematical ... more

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; Substitute in the equation and simplify.Since... more

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; Substitute and in the equation and simplify.... more

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; Substitute and in the equation and simplify.... more

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; First, calculate the value of .Hence, ; ... more

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; First, calculate the value of .Hence, . ; ... more

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-1 ; First, calculate the value of .Hence, ; ... more

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; Calculate ; Substitute, , and in the equation... more

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,, ; Find the value of and from the triangle as ... more

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; Find the value of  and  from the triangle. ; ... more

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; Find as the ratio of the base and the ... more

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; Find as the ratio of the base and the ... more

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; Use the cosine double-angle formula . ; more

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; Use the cosine double-angle formula and ... more

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; Use the cosine double-angle formula for the ... more

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; Use the cosine double-angle formula to reduce ... more

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  ; Given expression is Use the double-angle ... more

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  ; Given expression is Use the double-angle ... more

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The identity to prove is Using the square identity... more

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The identity to prove is .Use the double-angle ... more

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The identity to prove is Simplify the left-hand ... more

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The identity to prove is Simplify the left-hand ... more

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; Given expression is Use the double angle ... more

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; Given expression is Use the double angle ... more

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  ; Given expression is Rewrite this as and use ... more

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. ; Reduce the trigonometric expression using the... more

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  ; Given expression is Multiply and divide the ... more

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  ; Given expression is Multiply and divide the ... more

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; Given expression is . Write . Then use the ... more

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  ; Given expression is Write and then use the ... more

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; Given expression is . Use the double angle ... more

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; Given expression is Write this as Then, use the... more

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; Given expression is . Write . Then use the ... more

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; Given expression is Write and then use the ... more

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; The given function is . Use the reduction ... more

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; The given expression is Use the reduction ... more

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First, use the fact that and then to simplify ... more

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Use the identities and to simply the expression ... more

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Simplify the expression by using the trigonometric... more

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. ; Step 1: Consider the function and apply the ... more

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. ;  Expand the function by using the formula , ... more

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Use the trigonometric identities and to expand ... more

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Expand the square and use the identity to ... more

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Expand using the formula  Hence, it is shown that... more

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Expand using the formula . . Hence, it is proved ... more

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Expand the left hand side of the equality by using... more

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Expand the left hand side of equality using the ... more

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Step 1: Expand the left side of equality by using ... more

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. ; Expand it by using the product of cosines ... more

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. ; Expand the given expression by using the ... more

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. ; Use the product of cosine function to expand ... more

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. ; Use the sum of sine functions formula .Take ... more

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. ; Consider the left side of the equation and ... more

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. ; Interchange and in the given product to sum ... more

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The first method is to calculate the value by ... more

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The sum to product formula is used when it is ... more

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The product to sum formula is used when an ... more

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. ; Use the product to sum formula  by ... more

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. ; Use the product to sum of cosines formula   ... more

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. ; Use the formula for finding sum of products ... more

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. ; Using the formula   for finding sum of ... more

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. ; Using the formula   for finding sum of ... more

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. ; Using the formula   for finding sum of ... more

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. ; Using the formula   for finding the product of... more

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. ; Using the formula   for finding the product of... more

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; Using the formula   for finding the product of ... more

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; Using the formula   for finding the product, ... more

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. ; Using the formula   for finding the product of... more

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. ; Using the formula for finding the product, ... more

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. ; Using the formula for finding sum of products... more

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. ; Using the formula for finding the sum of ... more

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. ; Using the formula for finding the sum of ... more

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. ; Use the formula for finding the product using... more

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. ; Use the formula for finding the products by ... more

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. ; Use the formula   for finding the product by ... more

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. ; Use the formula for finding the product by ... more

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. ; Use the formula for the product of sines to ... more

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. ; The formula for the product of sines and ... more

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. ; The formula for the product of cosines to ... more

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. ; The formula for the product of sines and ... more

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. ; The formula of the difference of cosines to ... more

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. ; The formula of the difference of sines to ... more

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. ; The formula of the sum of cosines to convert ... more

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. ; The formula for the sum of sines to convert ... more

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Formulas for cosine applied over addition of two ... more

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Consider the left hand side expression of identity... more

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The formula for the product of sine and cosine to ... more

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Simplify the left-hand side of the equation using ... more

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Simplify the left-hand side expression by taking ... more

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Left hand side expression can be rewritten as . ... more

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Consider the left hand side expression of the ... more

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1.5081. ; The formula for the sum of cosines to ... more

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-0.0174. ; The difference of sines formula to ... more

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-0.2078. ; The difference of cosines formula to ... more

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-0.1561. ; The product of sine and cosine formula ... more

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-0.2410. ; The product of sines formula to convert... more

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To evaluate the left-hand side of the expression... more

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To evaluate the left-hand side of the expression... more

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To evaluate the left-hand side of the expression... more

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To evaluate the left-hand side of the expression... more

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To evaluate the left-hand side of the expression... more

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To evaluate the left-hand side of the expression... more

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To evaluate the expression , use the product to ... more

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To evaluate the expression , use the sum to ... more

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To evaluate the expression , use the sum to ... more

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To evaluate the expression , use the product to ... more

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To prove , take the left-hand side . Consider as... more

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To prove , take the left-hand side . Consider as... more

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To prove the identity, evaluate the left-hand side... more

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To prove the identity, evaluate the left-hand side... more

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To prove the identity, evaluate the left-hand side... more

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To prove the identity, evaluate the left-hand side... more

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To prove the identity, evaluate the left-hand side... more

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To prove the identity, evaluate the left hand side... more

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To prove the identity, evaluate the left-hand side... more

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To prove the identity, evaluate the left-hand side... more

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. ; The equation is rearranged to take the unknown... more

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. ; The value of lies in the interval 0 to , and ... more

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. ; Rewrite the for in terms of inverse function... more

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. ; Use the identity for substituting as in the... more

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. ; Solve the quadratic equation for , and then ... more

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No, a trigonometric equation may not have a ... more

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Yes, simplifying the equations in terms of one ... more

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When the term representing sine or cosine function... more

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. ; Isolate the function on to one side of the ... more

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 . ; Isolate the function on to one side of the ... more

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. ; The cosine function is isolated to one side of... more

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 . ; The cosine function is isolated to one side ... more

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. ; The tangent function is isolated to one side ... more

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. ; The tangent function is isolated to one side ... more

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. ; The cotangent function is isolated to one side... more

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. ; The sine function is isolated to one side of ... more

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. ; The cosine function is isolated to one side of... more

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. ;  The value of the cosine function is positive ... more

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. ; The value of cosine function is negative in ... more

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. ; The value of the sine function is negative in ... more

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, ; The given expression is .Use an algebraic ... more

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; The given expression is .Use anl agebraic ... more

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; The given expression is .An algebraic technique... more

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; The given expression is .An algebraic technique... more

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; The given expressionn is .An algebraic ... more

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; Solve the equation by isolating the sine term... more

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; The given expression is .An algebraic technique... more

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; The given expression is .The grouping technique... more

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; The given expression is .The grouping technique... more

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; The given expression is .Compare the equation ... more

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; The given expression is . Simplify the equation... more

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; The given expression is .The grouping and ... more

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; The given expression is .The algebraic ... more

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; The given expression is .The algebraic ... more

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; The given expression is .The equation is ... more

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; The given expression is .Compare the equation ... more

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; The given expression .The algebraic technique ... more

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; The given expression is .It is known that .The ... more

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; The given expression is .It is known that .The ... more

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; The given expression .It is known that .The ... more

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. ; Use the trigonometric formula   .Then, take ... more

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and . ; Simplify the equation using the double ... more

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. ; Use the formula of , and substitute it in the ... more

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. ; Use the formula of , and substitute it in the ... more

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. ; Use in the given equation. ; . ; Consider ... more

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. ; Factorize the equation by taking as the ... more

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. ; Let . The general solution of a quadratic ... more

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No solution. ; Consider that . The general ... more

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. ; Consider that . The general solution of a ... more

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. ; Let us consider as the solution of the ... more

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. ; Consider that . The general solution of a ... more

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. ; Consider that . The general solution of a ... more

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; Add on both side of equation then take as ... more

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No solution. ; Consider that . The general ... more

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; Use and replace by ;    ; Put Take common ... more

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 No solution.  ; According to the identity , for ... more

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; Use the identity then take common from both  ... more

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; Solve the equation using the identity and ... more

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; Add tern on each side of the equationUse the ... more

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; Add on each side of equation.Use then isolate ... more

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; Given ,But,So, the given equation becomes, ; ... more

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; Given,Divide both sides of the equation by ; ... more

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; Given,Divide both sides of the equation by ; ... more

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; Given ,Rearrange the equation,So this is a ... more

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; Given ,So this is a quadratic equation.Solve ... more

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; Given ,Rearrange the equation,So this is a ... more

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; Given ,But Substitute the value of from the ... more

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; Given ,But,Substitute the value of in equation... more

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; Given,But,Substitute this value in equation (1... more

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; Given,Rearrange the equation, ; So the value of... more

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; Given ,So this is a quadratic equation.Solve ... more

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; Given,So this is a quadratic equation. Solve ... more

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; Given,So this is a quadratic equation. Solve ... more

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No solutions. ; Given,So this is a quadratic ... more

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; Given,So this is a quadratic equation. Solve ... more

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; Given,So this is a quadratic equation. Solve ... more

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; Given,So this is a quadratic equation. Solve ... more

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; Set the calculator mode to radians.Press [] ... more

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; Set the calculator mode to radians.Press [] ... more

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; Set the calculator mode to radians.Press [] ... more

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; Take on both sides of the equation and solve ... more

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; Substitute in the equation such that, a ... more

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; Substitute in the equation such that, a ... more

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; Use the identity in the equation such that it... more

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; Use the identity to convert the equation in ... more

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; Assume that in the given equation and then ... more

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; It is given that .Use the identity to ... more

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  ; Assume that in the given equation and solve ... more

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; Apply the identity in the given equation and ... more

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; Apply the identity in the given equation and ... more

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No Solution. ; Apply the identity and in the ... more

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No solution. ; Apply the identity in the equation... more

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No solution ; Substitute and use to the equation... more

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; Apply and in the equation and solve for .By ... more

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; Substitute and   in the equation and solve ... more

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No solution. ; Apply the identity and substitute... more

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; Apply and substitute into the equation and ... more

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; Consider the situation as a right-angled ... more

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; Consider the situation as a right-angled ... more

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; Consider the situation as a right-angled ... more

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Approximately . ; Let be the angle at which the ... more

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; Let be the angle at which astronaut looks at ... more

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Approximately . ; Let   be the angle at which the ... more

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Approximately . ; Let   be the angle at which the ... more

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Approximately ; Let be the angle of elevation of ... more

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Approximately ; Let be the angle of elevation of ... more

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Approximately ; Let the spotlight be at an angle ... more

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Approximately ; Let be the angle made by the light... more

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Approximately ; Let be the angle made by the feet ... more

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Approximately ; Let be the angle made by the feet ... more

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Approximately ; Let be the angle made by the ... more

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; Simplify the equation by using to obtain the ... more

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; Simplify the equation to obtain the solutions ... more

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; Simplify the equation to obtain the solution ... more

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. ; Simplify the equation by rewriting as , and ... more

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. ; Simplify the equation by factorizing the ... more

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; Simplify the equation to obtain the solution ... more

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1 ; To simplify the expression , use the ... more

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. ; To simplify the expression , use the ... more

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Identities are equivalent. ; For the equation , ... more

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Yes ; Simplify the left side of the equation by ... more

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; Simplify the value by breaking the angle in ... more

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; Break the angle of the function into the sum ... more

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; Simplify the given function with the help of ... more

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; Simplify the expression by applying the ... more

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Consider the left side of the equation. Split the ... more

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Consider the left side of the equation. Split the ... more

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; Simplify the expression with the help of ... more

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; Substitute and and simplify the given ... more

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; Substitute the values and and simplify the ... more

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; Find the value of by sybstituting in the ... more

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; Assume a right angle triangle with base and ... more

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; Find the value of . ; Find the value of with ... more

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; Find the value of . ; Find the value of with ... more

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; Find the values of the height from the ... more

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; Find the value of from the half angle formula... more

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Substitute and and simplify the left side of the... more

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Consider the left side of the equation. Substitute... more

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; Use identity   to simplify the expression ; ... more

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; Simplify the expression by the use of the ... more

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; Product to sum is given by,  ; more

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; Product to sum is given by,  ; more

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0.809 ; Product to sum is given by,    ; more

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; Sum to product is given by,      ; more

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0 ; Sum to product is given by,  ; more

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; Product to sum is given by,     ; more

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; Product to sum is given by,   ; more

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; Sum to product  is given by,    ; more

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; Sum to product is given by,  ; more

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; Solving the equation gives, ; more

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; Solving the equation gives,  ; more

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; Solving the equation gives,  ; more

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; Solving the equation gives,  ; more

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; Solving the equation gives,  ; more

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; Solving the equation gives,  ; more

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; Solving the equation gives,  ; more

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; Solving the equation gives,  ; more

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; Solving the equation gives,  ; more

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From the graph it can be inferred that the ... more

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As it is seen in the graph,there is no possible ... more

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; To simplify the the equation use the formula... more

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; To simplify the the equation use the Even odd ... more

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; To simplify the the equation substitute  and ... more

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1 ; To simplify the the equation substitute and ... more

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  ; We know that the angle is in the Second ... more

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  ; We know that the angle is in the Second ... more

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; firstly convert into for given equation   ; ... more

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; substitute the value in equation   ; more

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; use sum formula of cosine, ; more

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; use sum formula of tangent ; more

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   ; using the formula in equation and simplify ... more

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; use half angle identity for simplify the ... more

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; Substitute the value of in equation ; now ... more

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; Substitute the value of in equation ; now ... more

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; Apply the formula of cosine in equation ; more

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; Take the common from ; more

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; use the identity to convert the sum  into ... more

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; Use the identity of for equation substitute... more

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; Use the identity in equation of ; more

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; First of all take the constant value in right ... more

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Solution: . ; Take all the terms on one side of ... more

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. ; On the interval , the equation has the ... more

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. ; First, simplify the equation to isolate the ... more

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. ; Cosine is negative and sine is positive in the... more

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. ; is in the fourth quadrant, hence lies in the... more

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. ; Use the double-angle formula, , to simplify ... more

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Use the identity to prove the identity. ; Use the... more

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Use the identities, , .. ; Use the identities... more

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Use the identities , to simplify the expression... more

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; Using CAS, the cosine function that fits the ... more

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; Displacement function is . A sine function take... more

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; Height of the building, . Distance from the ... more

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1. Slow Oscillation: Amplitude: 4Period: Frequency... more

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Jan, May ; Since the average monthly snowfall is ... more

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  ; At time, , intial amplitude . For damped ... more

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