a. Answer sin 3 π 4 = 1 2 Explanation Given information: sin 3 π 4 Formula used: Finding value by using reference angle of given angle. Reference angle for different quadrant: 1 s t quadrant: θ r e f = θ 2 n d quadrant: θ r e f = π − θ 3 r d quadrant: θ r e f = θ − π 4 t h quadrant: θ r e f = 2 π − θ As angle lies in 2 n d quadrant. So, θ r e f = π − θ θ r e f = π − 3 π 4 θ r e f = 4 π 4 − 3 π 4 θ r e f = π 4 As sin θ is positive in 2 n d quadrant, So sin 3 π 4 = sin π 4 = 1 2 b. To determine Exact value Trigonometric function at the given real number. Answer sin ( 5 π 4 ) = − 1 2 Explanation Given information: sin ( 5 π 4 ) Formula used: Finding value by using reference angle of given angle. Reference angle for different quadrant: 1 s t quadrant: θ r e f = θ 2 n d quadrant: θ r e f = π − θ 3 r d quadrant: θ r e f = θ − π 4 t h quadrant: θ r e f = 2 π − θ As angle lies in 3 r d quadrant. So, θ r e f = θ − π θ r e f = 5 π 4 − π θ r e f = 5 π 4 − 4 π 4 θ r e f = π 4 As sin θ is negative in 3 r d quadrant, So cos 5 π 4 = cos π 4 = − 1 2 c. To determine Exact value Trigonometric function at the given real number. Answer sin 7 π 4 = − 1 2 Explanation Given information: sin 7 π 4 Formula used: Finding value by using reference angle of given angle. Reference angle for different quadrant: 1 s t quadrant: θ r e f = θ 2 n d quadrant: θ r e f = π − θ 3 r d quadrant: θ r e f = θ − π 4 t h quadrant: θ r e f = 2 π − θ As angle lies in 4 t h quadrant. So, θ r e f = 2 π − θ θ r e f = 2 π − 7 π 4 θ r e f = 8 π 4 − 7 π 4 θ r e f = π 4 As sin θ is negative in 4 t h quadrant, So sin 7 π 4 = sin π 4 = − 1 2

Precalculus: Mathematics for Calculus - 6th Edition

6th Edition

ISBN: 9780840068071

Author: Stewart, James, Redlin, Lothar, Watson, Saleem

Publisher: Cengage Learning

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Concept explainers

Fundamentals of Trigonometric Identities

Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.

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Question

Chapter 5.2, Problem 10E

To determine

Exact value Trigonometric function at the given real number.

Expert Solution

Answer to Problem 10E

sin3π4=12

Explanation of Solution

Given information:

sin3π4

Formula used:

Finding value by using reference angle of given angle.

Reference angle for different quadrant:

1^s^t quadrant: θref=θ

2ⁿ^d quadrant: θref=π−θ

3^r^d quadrant: θref=θ−π

4^t^h quadrant: θref=2π−θ

As angle lies in 2ⁿ^d quadrant. So,

θref=π−θθref=π−3π4θref=4π4−3π4θref=π4

As sinθ is positive in 2ⁿ^d quadrant, So

sin3π4=sinπ4=12

To determine

Exact value Trigonometric function at the given real number.

Expert Solution

Answer to Problem 10E

sin(5π4)=−12

Explanation of Solution

Given information:

sin(5π4)

Formula used:

Finding value by using reference angle of given angle.

Reference angle for different quadrant:

1^s^t quadrant: θref=θ

2ⁿ^d quadrant: θref=π−θ

3^r^d quadrant: θref=θ−π

4^t^h quadrant: θref=2π−θ

As angle lies in 3^r^d quadrant. So,

θref=θ−πθref=5π4−πθref=5π4−4π4θref=π4

As sinθ is negative in 3^r^d quadrant, So

cos5π4=cosπ4=−12

To determine

Exact value Trigonometric function at the given real number.

Expert Solution

Answer to Problem 10E

sin7π4=−12

Explanation of Solution

Given information:

sin7π4

Formula used:

Finding value by using reference angle of given angle.

Reference angle for different quadrant:

1^s^t quadrant: θref=θ

2ⁿ^d quadrant: θref=π−θ

3^r^d quadrant: θref=θ−π

4^t^h quadrant: θref=2π−θ

As angle lies in 4^t^h quadrant. So,

θref=2π−θθref=2π−7π4θref=8π4−7π4θref=π4

As sinθ is negative in 4^t^h quadrant, So

sin7π4=sinπ4=−12

Chapter 5.2, Problem 9E

Chapter 5.2, Problem 11E

Chapter 5 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

Ch. 5.1 - Prob. 1E Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E Ch. 5.1 - Prob. 6E Ch. 5.1 - Prob. 7E Ch. 5.1 - Prob. 8E Ch. 5.1 - Prob. 9E Ch. 5.1 - Prob. 10E

Ch. 5.1 - Prob. 11E Ch. 5.1 - Prob. 12E Ch. 5.1 - Prob. 13E Ch. 5.1 - Prob. 14E Ch. 5.1 - Prob. 15E Ch. 5.1 - Prob. 16E Ch. 5.1 - Prob. 17E Ch. 5.1 - Prob. 18E Ch. 5.1 - Prob. 19E Ch. 5.1 - Prob. 20E Ch. 5.1 - Prob. 21E Ch. 5.1 - Prob. 22E Ch. 5.1 - Prob. 23E Ch. 5.1 - Prob. 24E Ch. 5.1 - Prob. 25E Ch. 5.1 - Prob. 26E Ch. 5.1 - Prob. 27E Ch. 5.1 - Prob. 28E Ch. 5.1 - Prob. 29E Ch. 5.1 - Prob. 30E Ch. 5.1 - Prob. 31E Ch. 5.1 - Prob. 32E Ch. 5.1 - Prob. 33E Ch. 5.1 - Prob. 34E Ch. 5.1 - Prob. 35E Ch. 5.1 - Prob. 36E Ch. 5.1 - Prob. 37E Ch. 5.1 - Prob. 38E Ch. 5.1 - Prob. 39E Ch. 5.1 - Prob. 40E Ch. 5.1 - Prob. 41E Ch. 5.1 - Prob. 42E Ch. 5.1 - Prob. 43E Ch. 5.1 - Prob. 44E Ch. 5.1 - Prob. 45E Ch. 5.1 - Prob. 46E Ch. 5.1 - Prob. 47E Ch. 5.1 - Prob. 48E Ch. 5.1 - Prob. 49E Ch. 5.1 - Prob. 50E Ch. 5.1 - Prob. 51E Ch. 5.1 - Prob. 52E Ch. 5.1 - Prob. 53E Ch. 5.1 - Prob. 54E Ch. 5.1 - Prob. 55E Ch. 5.1 - Prob. 56E Ch. 5.1 - Prob. 57E Ch. 5.1 - Prob. 58E Ch. 5.2 - Let P(x, y) be the terminal point on the unit...Ch. 5.2 - 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The cancellation property sin1(sin x) = x is valid...Ch. 5.5 - Prob. 3E Ch. 5.5 - Prob. 4E Ch. 5.5 - Prob. 5E Ch. 5.5 - Prob. 6E Ch. 5.5 - Prob. 7E Ch. 5.5 - Prob. 8E Ch. 5.5 - Prob. 9E Ch. 5.5 - Prob. 10E Ch. 5.5 - Prob. 11E Ch. 5.5 - Prob. 12E Ch. 5.5 - Prob. 13E Ch. 5.5 - Prob. 14E Ch. 5.5 - Prob. 15E Ch. 5.5 - Prob. 16E Ch. 5.5 - Prob. 17E Ch. 5.5 - Prob. 18E Ch. 5.5 - Prob. 19E Ch. 5.5 - Prob. 20E Ch. 5.5 - Prob. 21E Ch. 5.5 - Prob. 22E Ch. 5.5 - Prob. 23E Ch. 5.5 - Prob. 24E Ch. 5.5 - Prob. 25E Ch. 5.5 - Prob. 26E Ch. 5.5 - Prob. 27E Ch. 5.5 - Prob. 28E Ch. 5.5 - Prob. 29E Ch. 5.5 - Prob. 30E Ch. 5.5 - Prob. 31E Ch. 5.5 - Prob. 32E Ch. 5.5 - Prob. 33E Ch. 5.5 - Prob. 34E Ch. 5.5 - Prob. 35E Ch. 5.5 - Prob. 36E Ch. 5.5 - Prob. 37E Ch. 5.5 - Prob. 38E Ch. 5.5 - Prob. 39E Ch. 5.5 - Prob. 40E Ch. 5.5 - Prob. 41E Ch. 5.5 - Prob. 42E Ch. 5.5 - Prob. 43E Ch. 5.5 - Prob. 44E Ch. 5.5 - Prob. 45E Ch. 5.5 - Prob. 46E Ch. 5.5 - Prob. 47E Ch. 5.6 - For an object in simple harmonic motion with...Ch. 5.6 - For an object in damped harmonic motion with...Ch. 5.6 - Prob. 3E Ch. 5.6 - Prob. 4E Ch. 5.6 - Prob. 5E Ch. 5.6 - Prob. 6E Ch. 5.6 - Prob. 7E Ch. 5.6 - Prob. 8E Ch. 5.6 - Prob. 9E Ch. 5.6 - Prob. 10E Ch. 5.6 - Prob. 11E Ch. 5.6 - Prob. 12E Ch. 5.6 - Prob. 13E Ch. 5.6 - Prob. 14E Ch. 5.6 - Prob. 15E Ch. 5.6 - Prob. 16E Ch. 5.6 - Prob. 17E Ch. 5.6 - Prob. 18E Ch. 5.6 - Prob. 19E Ch. 5.6 - Prob. 20E Ch. 5.6 - Prob. 21E Ch. 5.6 - Prob. 22E Ch. 5.6 - Prob. 23E Ch. 5.6 - Prob. 24E Ch. 5.6 - Prob. 25E Ch. 5.6 - Prob. 26E Ch. 5.6 - A Bobbing Cork A cork floating in a lake is...Ch. 5.6 - FM Radio Signals The carrier wave for an FM radio...Ch. 5.6 - Blood Pressure Each time your heart beats, your...Ch. 5.6 - Predator Population Model In a predator/prey...Ch. 5.6 - Prob. 31E Ch. 5.6 - Tides The graph shows the variation of the water...Ch. 5.6 - Tides The Bay of Fundy in Nova Scotia has the...Ch. 5.6 - Prob. 34E Ch. 5.6 - Prob. 35E Ch. 5.6 - Prob. 36E Ch. 5.6 - Ferris Wheel A Ferris wheel has a radius of 10 m,...Ch. 5.6 - Cock Pendulum The pendulum in a grandfather clock...Ch. 5.6 - Variable Stars The variable star Zeta Gemini has a...Ch. 5.6 - Variable Stars Astronomers believe that the radius...Ch. 5.6 - Biological Clocks Circadian rhythms are biological...Ch. 5.6 - Electric Generator The armature in an electric...Ch. 5.6 - Electric Generator The graph shows an oscilloscope...Ch. 5.6 - Doppler Effect When a car with its horn blowing...Ch. 5.6 - Motion of a Building A strong gust of wind strikes...Ch. 5.6 - Shock Absorber When a car hits a certain bump on...Ch. 5.6 - Tuning Fork A tuning fork is struck and oscillates...Ch. 5.6 - Guitar String A guitar string is pulled at point P...Ch. 5 - Prob. 1RCC Ch. 5 - Prob. 2RCC Ch. 5 - Prob. 3RCC Ch. 5 - Prob. 4RCC Ch. 5 - Prob. 5RCC Ch. 5 - Prob. 6RCC Ch. 5 - Prob. 7RCC Ch. 5 - Prob. 8RCC Ch. 5 - Prob. 9RCC Ch. 5 - Prob. 10RCC Ch. 5 - Prob. 11RCC Ch. 5 - Prob. 12RCC Ch. 5 - Prob. 1RE Ch. 5 - Prob. 2RE Ch. 5 - Prob. 3RE Ch. 5 - Prob. 4RE Ch. 5 - Prob. 5RE Ch. 5 - 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Fundamental Trigonometric Identities: Reciprocal, Quotient, and Pythagorean Identities; Author: Mathispower4u;https://www.youtube.com/watch?v=OmJ5fxyXrfg;License: Standard YouTube License, CC-BY