Algebra and Trigonometry: Real Mathematics, Real People (7th Edition)

Chapter 8

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: ; : units; : units ; The sum of all the angles... more

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: ; : feet ; Find the angle after tilting the ... more

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: ; : ;: unit ; Draw the triangle with given sides... more

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Entire Proof Answer:Use the Law of Sines in ... more

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; ; ; ;    ; Use the formula to find the height ... more

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square meters  ; Draw the triangle with given ... more

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; Consider triangle ABC has sides a, b, c, and ... more

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Oblique ; An oblique triangle is a triangle that ... more

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; The rule is called the sine rule. Here the ... more

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; The formulae , and are used to determine the ... more

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Angles ; Any triangle with two angles and one side... more

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The two cases given are two angles and one side, ... more

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Yes, it is always true that the longest side of a ... more

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; The sum of all the angles in a triangle is . ... more

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; The sum of all the angles in a triangle is . ... more

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; The sum of all the angles in a triangle is . ... more

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; The sum of all the angles in a triangle is . ... more

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; The sum of all the angles in a triangle is . ... more

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; The sum of all the angles in a triangle is . ... more

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; Use the law of sines in a reciprocal form to ... more

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; Use the law of sine in reciprocal form to find... more

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; The sum of all the angles in a triangle is . ... more

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; The sum of all the angles in a triangle is . ... more

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; Use the law of sine in reciprocal form to find ... more

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; Use the law of sine in reciprocal form to find ... more

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; The sum of all the angles in a triangle is . ... more

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; The sum of all the angles in a triangle is . ... more

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; The sum of all the angles in a triangle is . ... more

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; The sum of all the angles in a triangle is . ... more

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; Use the law of sine in reciprocal form to find... more

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; Use the law of sine in reciprocal form to find... more

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No solution ; Use the law of sine in reciprocal ... more

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No solution ; In any triangle, the greatest angle ... more

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No solution ; In any triangle, the greatest angle ... more

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,when , when ; In the given triangle one angle is ... more

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,when , when ; It is given one angle is acute, , ... more

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No solution ; Consider the sides and angle for a ... more

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; Use the formula of area for an oblique triangle... more

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; The area of any triangle is one-half of the ... more

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; The area of any triangle is one-half of the ... more

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; The area of any triangle is one-half of the ... more

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; The area of any triangle is one-half of the ... more

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; The area of any triangle is one-half of the ... more

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24.12 meters ; Sketch the visual diagram of the ... more

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77 meters ; Sketch the visual diagram of the given... more

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; Sketch the visual diagram of the given problem... more

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: ; : ; Redraw the diagram given in the problem. ... more

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; Refer to the given diagram and substitute 17 ... more

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; Redraw the given visual representation of the ... more

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False, there is a need of at least one angle when ... more

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True, as one of the angle is more than , the other... more

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False, for a unique solution the condition must ... more

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YesSince two angles are given, use the triangle ... more

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; Use the quotient property of Tangent function ... more

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; Use the reciprocal property of Cotangent ... more

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; Write the given expression as the sum or ... more

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; Write the given expression as the sum or ... more

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; Write the given expression as the sum or ... more

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; Write the given expression as the sum or ... more

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Pythagoras theoremThe Pythagoras theorem is a ... more

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On solving side , any of the angle or can be ... more

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Yes, the given formulas can be used to find the ... more

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; The values of the sides of the triangle are... more

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; The values of the sides of the triangle are , ... more

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feet ; Plot the figure for the given information.... more

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; Substitute 30 for , 56 for , and 40 for into ... more

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19.90 square feet ; Use Heron’s formula to ... more

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; For a triangle with angles A, B, and C, ... more

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; The area of a triangle, right-angled at C, with... more

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No ; The ASA, that is, two angles and included ... more

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Yes ; The SAS, that is, two sides and their ... more

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Yes ; The SSS, that is, three sides of the ... more

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No ; The SSA, that is, two sides and an angle ... more

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; The values of the sides of the triangle are... more

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; The values of the sides of the triangle are... more

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; The values of the sides of the triangle are , ... more

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; The values of the sides of the triangle are... more

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; The values of the sides of the triangle are , ... more

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; The values of the sides of the triangle are , ... more

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; The values of the sides of the triangle are... more

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; The values of the sides of the triangle are... more

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; The values of the sides of the triangle are , ... more

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; The values of the sides of the triangle are , ... more

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; The values of the sides of the triangle are , ... more

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; The values of the sides of the triangle are , ... more

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; The values of the sides of the triangle are... more

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; The values of the sides of the triangle are... more

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; The values of the sides of the triangle are... more

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; The values of the sides of the triangle are , ... more

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; The sides of the triangle are and . The given ... more

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; The sides of the triangle are and . The given ... more

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 48 ; From the given table,the values of sides a, ... more

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 2535 ; From the given table,The values of sides a... more

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 101420 ; From the given table,The values of sides... more

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 406080 ; From the given table,The values of sides... more

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; Plot the figure for the given information.Apply... more

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; Plot the figure for the given information.Apply... more

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Law of Cosines ; From the given triangle ABC, the ... more

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Law of Sine ; From the given triangle ABC, the ... more

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Law of Cosines ; The values of sides a, c, and ... more

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Law of Cosines ; The values of sides a, b, and ... more

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Law of SineNo triangle will be formed ; The values... more

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Law of Cosines ; The values of sides a, b, and c ... more

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Law of Sine ; The values of side c and the angles ... more

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Law of Sine ; The values of sides b, c, and angle ... more

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104.57 square units ; Use Heron’s formula to ... more

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385.70 square units ; Use Heron's formula to ... more

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19.81 square units ; Use Heron’s formula to ... more

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43.52 square units ; Use Heron’s formula to ... more

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0.27 square units ; Use Heron’s formula to ... more

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2.57 square units ; Use Heron’s formula to ... more

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0.18 square units ; Use Heron’s formula to ... more

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0.11 square units ; Use Heron’s formula to ... more

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483.40 meters ; Refer to the given figure.Assume ... more

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; Refer to the given figure.Substitute 2 for a , ... more

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miles ; Assume that the ship that leaves the port ... more

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Approx ; The length of the sides a,b, and c of the... more

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feet5 feetfeet ; Substitute the value of height (h... more

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feetfeet ; The sidehill angle of the tower is . As... more

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feet ; Draw the figure by using the given ... more

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Approx ; The length of the sides a,b, and c of the... more

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Approx ; Compute the height of the parallelogram ... more

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True The Law of cosines is used to find the side ... more

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False ; The area of a triangle with sides of ... more

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Result obtained from both the methods is the same.... more

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; Since function has an inverse in the interval... more

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; Since function has an inverse in the interval... more

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; Since function has an inverse in the interval... more

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; Since function has an inverse in the interval... more

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Equivalent Vectors ; Calculate the magnitude of ... more

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; Consider the given points and ; vector can be... more

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; Divide by its magnitude to calculate the unit ... more

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; Calculate the vector using vector component ... more

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; Substitute the value of vectors and and ... more

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; As the plane is descending at an angle of with... more

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2,405 pounds ; Let be the combined weight of the ... more

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miles per hour; ; Draw the figure with the help of... more

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Directed line segment ; Any quantity that requires... more

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InitialTerminal ; The directed line segment will ... more

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Magnitude ; If and are two vectors .The ... more

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Vector ; Two lines are equivalent to each other if... more

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Standard position ; A vector is the collection of... more

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Multiplication ; Scalar multiplication and vector ... more

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Resultant ; The result obtained by adding two ... more

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Linear combinationHorizontalVertical ; can be ... more

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Magnitude and direction ; Two directed line ... more

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Unit vector ; Any vector that has magnitude equal ... more

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Entire proof Answer: Represent and in component ... more

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Entire proof Answer: Represent and in component ... more

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; Consider the given points and . Vector is ... more

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; Consider the given points and . Vector is ... more

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; Consider the given points and . Vector is ... more

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; Consider the given points and . Vector is ... more

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; Consider the given points and . Vector is ... more

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; Consider the given points and . Vector is ... more

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; Consider the given points and . Vector is ... more

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; Consider the given points and . Vector is ... more

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; Consider the given points and . Vector is ... more

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; Consider the given points and . Vector is ... more

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; Consider the given points and . Vector is ... more

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; Consider the given points and . Vector is ... more

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T product of the vector, and scalar is a vector ... more

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T product of the vector and scalar three is a ... more

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To add the vectors and , position them so that ... more

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The vector, will have a direction directly ... more

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The vector, will have a direction directly ... more

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The vector, will have the same direction as that ... more

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The magnitude of vector is two times the ... more

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The magnitude of vector is three times the ... more

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Vector have the same direction as that of vector... more

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Vector has the same direction as that of vector ... more

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Vector has the same direction as that of vector ... more

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The vector will have the same direction as that ... more

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; To find the vector , apply the parallelogram ... more

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  ; The vector is generated by joining the ... more

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; The vector is generated by joining the ... more

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  ; The vector is generated by joining the ... more

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; In order to find the unit vector in the ... more

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; In order to find the unit vector in the ... more

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; In order to find the unit vector in the ... more

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; In order to find the unit vector in the ... more

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; In order to find the unit vector in the ... more

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; In order to find the unit vector in the ... more

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; In order to find the unit vector in the ... more

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; In order to find the unit vector in the ... more

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; Divide vector by its magnitude to find the ... more

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; Divide vector by its magnitude to find the ... more

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; Divide vector by its magnitude to find the ... more

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; Divide vector by its magnitude to find the ... more

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; Divide vector by its magnitude to find the ... more

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; Divide vector by its magnitude to find the ... more

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; Divide vector by its magnitude to find the ... more

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; Divide vector by its magnitude to find the ... more

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; Subtract initial point from terminal point to ... more

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; Subtract initial point from terminal point to ... more

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; Subtract initial point from terminal point to ... more

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; Subtract initial point from terminal point to ... more

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; In order to find the component form of vector... more

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; In order to find the component form of vector... more

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; Substitute the value of vector and in and ... more

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; Substitute vectors and in vector . ; Add ... more

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; Substitute vectors and in given vector. ; Add... more

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; Substitute vectors and in given vector. ; Add... more

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; Substitute the values of and . ;   ; Calculate... more

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; Substitute the values of and . ; Calculate the... more

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; Compute the magnitude of the given vector. ; ... more

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; Evaluate the magnitude of the given vector. ; ... more

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; Calculate the magnitude of the given vector. ; ... more

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; Calculate the magnitude of the given vector. ; ... more

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; Components of vector having is calculated as, ... more

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; Components of vector having is calculated as,... more

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; Components of vector having is calculated as,... more

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; As components of vector is calculated as, ... more

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; As , components of vector is calculated as,... more

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; Calculate components of vector . ; The ... more

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; As vector is in direction of , find the unit ... more

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; As vector is in the direction of , find the ... more

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; Find the components of vector and . ;   ; ... more

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; Find the components of vector and . ; Write ... more

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; Calculate the components of vector and . ; ... more

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; Calculate the components of vector and... more

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Approx ; Assume Force1() is along x-axis and ... more

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; Find the vector using the vectors given. ; ... more

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; Write the given vectors in terms of the angle ... more

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; Write the given vectors in terms of the angle ... more

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; The force 1 is along the positive -axis and ... more

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; Assume Force1() is along x-axis and Force2() is... more

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Approx Approx ; Let the horizontal component be ... more

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Approx Approx ; Let the horizontal component be ... more

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poundspounds ; Draw the free body diagram ... more

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Approx poundsApprox pounds ; Draw the free body ... more

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km/hr; ; Draw a free body diagram of airplane ... more

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TrueAs a vector has only two components, magnitude... more

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TrueRearranging the equation of unit vector in ... more

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TrueAs each component is zero, equating the sum of... more

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TrueEquating the magnitude of vector with unity ... more

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Entire Proof Answer: Consider the scalars, as the... more

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Follow the listed steps to perform vector ... more

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:;: ; In reference to the graph given in the ... more

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: ; : ; In reference to the graph given in the ... more

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; Using the rules of addition and subtraction of ... more

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; Use the rules of addition and subtraction of ... more

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,   ; Simplify the given expression by using the ... more

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, ; Simplify the given expression by using the ... more

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; Group decimal numbers and power terms with base... more

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; Group decimal numbers and power terms with base... more

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; Equate to zero to solve for . ;   ; Take on ... more

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; Equate to zero to solve for . ; Take on both ... more

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; Subtract 4 from both sides in the given ... more

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; Subtract 18 from both sides in the given ... more

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; Substitute for in the given expression. ; Use... more

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; Substitute for in the given expression. ; ... more

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; Write the dot product of the given vectors and... more

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; Find the dot product of the vectors and . ; ... more

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Orthogonal ; Two vectors are orthogonal when their... more

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; Calculate the dot product of and . ; Calculate... more

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pounds ; Write the gravitational force that acts ... more

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foot-pounds ; Let be the force applied on the ... more

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Yes ; Dot products are distributive in nature.... more

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0 ; The dot product of two vectors is given as .... more

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Scalar ; In the dot product of vectors, the ... more

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; The dot product of two vectors is given as . ... more

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; Let be a vector whose projection onto is . ... more

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or ; Work done by a constant force , as its point ... more

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0 ; Apply the formula of dot product and simplify... more

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; Write the dot product of the given vectors and... more

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14 ; Write the given vectors and in component ... more

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36 ; Write and in component form. ; Substitute ... more

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8Scalar ; Calculate the dot product of with ... more

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 Scalar ; Calculate the dot product of and . ; ... more

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 Scalar ; Calculate the vector . ; Calculate the ... more

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Scalar ; Calculate the dot product of and . ; The... more

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; Vector  ; Multiply by 3 to find . ;   ; Use the... more

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Vector ; Multiply by 2 to find . ; Use the ... more

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13 ; Use the formula to calculate the magnitude ... more

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17 ; Use the formula to calculate the magnitude ... more

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; Use the formula to calculate the magnitude of... more

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; Write in terms of its components. ; Use the ... more

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4 ; Write in terms of its components. ; Use the ... more

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9 ; Write in terms of its components. ; Use the ... more

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; Find the dot product of the vectors and . ; ... more

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; Find the dot product of the vectors and . ; ... more

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; Write the given vectors and in component form... more

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; Write the given vectors and in terms of its ... more

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; Write the given vectors and in terms of its ... more

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; Write the given vectors and in terms of its ... more

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; Substitute the values of ,,, and  into the ... more

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; Substitute the values of , , , and into the ... more

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; Sketch the graph of the given vectors and in ... more

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; Sketch the graph of the given vectors and in ... more

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; Sketch the graph of the given vectors and in ... more

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; Sketch the graph of the given vectors and in ... more

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; ; ; Let , , and, , write these in terms of three... more

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; ; ; Let , , and,  , write these in terms of ... more

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;; ; Let , , and,  , write these in terms of three... more

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;; ; Let , , and .Write the vertices , , and of ... more

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; Substitute the given magnitudes of and and ... more

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24 ; Substitute the given magnitudes of and and ... more

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; Substitute the given magnitudes of and and ... more

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; To find the dot product , rewrite the formula ... more

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and are orthogonal. ; Calculate the dot product ... more

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and are not orthogonal. ; If the dot product of ... more

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and are not orthogonal. ; Write the horizontal ... more

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and are orthogonal. ; Write the horizontal and ... more

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Calculate the dot product of and .The dot product... more

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Calculate the dot product of and .The dot product... more

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Calculate the dot product of and .The dot product... more

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Calculate the dot product of and .The dot product... more

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; Calculate the dot product of vectors,   and... more

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; Obtain the dot product of vectors, and . ; The... more

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; Calculate the dot product of vectors, and . ; ... more

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; Calculate the dot product of vectors, and . ; ... more

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; Calculate the dot product of vectors, and . ; ... more

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; Calculate the dot product of vectors, and . ; ... more

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; Calculate the dot product of and . ; Calculate... more

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; Calculate the dot product of and . ; Calculate... more

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; ; Calculate the dot product of and . ; Use the ... more

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; ; Calculate the dot product of and . ; Use the ... more

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;    Vectors and are in the same direction, so ... more

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; The given vectors and are parallel to each ... more

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; The vectors and are perpendicular to each ... more

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.  ; The vectors and are perpendicular to each ... more

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; ; Let and are the two vectors in opposite ... more

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; ; Let and be two vectors in opposite direction... more

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;   ; Let and be two vectors in opposite ... more

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; ; Let and be two vectors in opposite direction... more

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32 ; Calculate the distance vector . ; Apply the ... more

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14 ; Calculate the distance vector . ; The vector... more

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5318 pounds ; Write the gravitational force vector... more

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ft-lb ; The given situation can be represented in ... more

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foot-pounds ; The given situation can be ... more

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foot-pounds ; The given situation can be ... more

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True, The dot product of and is 0. ; Calculate ... more

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False, the work done is a scalar quantity not a ... more

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Orthogonal ; Calculate the dot product of the ... more

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The result of the dot product is a scalar, not a ... more

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; Let the unit vector u be .Calculate .If u is a ... more

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Entire Proof Answer:Let  the sides of the rhombus ... more

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Entire Proof Answer: Assume that the vectors  and... more

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The vector u is orthogonal to v and w, so the dot ... more

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Draw the unit vector u which makes an angle with ... more

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Draw the unit vector u which makes an angle with ... more

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The function is the horizontal shift of units to... more

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The function is the reflection of with respect ... more

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The function is the upward shift of  6 units  of ... more

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The horizontal shrink of the function is . ; Let ... more

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; Apply distributive property to simplify the ... more

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; Apply distributive property to simplify the ... more

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; Apply the identity, to simplify the given ... more

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; Apply the identity, to simplify the given ... more

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; Multiply the numerator and denominator by the ... more

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; Multiply the numerator and denominator by the ... more

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Absolute value: ; The absolute value of any ... more

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; Write the complex number in the form of . ; ... more

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; Find the absolute value of . ; Substitute as ... more

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; The value of is and that of is . Substitute ... more

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; Write the formula of the product of two complex... more

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; Write the formula of the product of two complex... more

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; Write the formula of the division of two ... more

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-4 ; Write the complex number in trigonometric ... more

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  ; Write the number 1 in trigonometric form by ... more

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; The absolute value of the complex number is... more

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There is a difference between the value of used ... more

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The value of increases in multiples of   as the ... more

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; Solve the equation .Write in trigonometric ... more

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Set the value of Tstep as , the resulting graph ... more

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Absolute value ; a in the complex number is ... more

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DeMoivre’s ; DeMoivre’s theorem states that for ... more

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nth root ; A complex number is the nth root of a ... more

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; The complex number is written in the form ... more

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In the trigonometric form , is the distance ... more

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In the trigonometric form , is the angle measured... more

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; Find the absolute value r of the function .  ; ... more

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; Find the absolute value r of the function . The... more

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; Find the absolute value r of the function . The... more

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; Find the absolute value r of the function . The... more

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; Find the absolute value r of the function . The... more

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; Find the absolute value r of the function . The... more

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; Find the absolute value r of the function . The... more

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; Find the absolute value r of the function . The... more

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; Write the complex number given as . Find the ... more

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; Look at the graph and write the given complex ... more

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; Look at the graph and get the complex number ... more

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; Look at the graph and get the complex number ... more

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; Look at the graph and get the complex number ... more

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