Textbook Question
In Exercises 37โ38, a point on the terminal side of angle ฮธ is given. Find the exact value of each of the six trigonometric functions of ฮธ, or state that the function is undefined.
(0, -1)
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 1, Problem 39
In Exercises 33โ42, let sin t = a, cos t = b, and tan t = c. Write each expression in terms of a, b, and c.
sin(-t - 2๐) - cos(-t - 4๐) - tan(-t - ๐)
Verified step by step guidance1
Recall the even-odd properties of trigonometric functions: \(\sin(-x) = -\sin x\), \(\cos(-x) = \cos x\), and \(\tan(-x) = -\tan x\).
Use the periodicity of the functions: \(\sin(x - 2\pi) = \sin x\), \(\cos(x - 4\pi) = \cos x\), and \(\tan(x - \pi) = \tan x\).
Apply these properties to each term in the expression:
- For \(\sin(-t - 2\pi)\), rewrite as \(\sin(-(t + 2\pi)) = -\sin(t + 2\pi) = -\sin t = -a\).
- For \(\cos(-t - 4\pi)\), rewrite as \(\cos(-(t + 4\pi)) = \cos(t + 4\pi) = \cos t = b\).
- For \(\tan(-t - \pi)\), rewrite as \(\tan(-(t + \pi)) = -\tan(t + \pi) = -\tan t = -c\).
Substitute the simplified terms back into the original expression: \(\sin(-t - 2\pi) - \cos(-t - 4\pi) - \tan(-t - \pi) = (-a) - b - (-c)\).
Simplify the expression by combining like terms: \(-a - b + c\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Function Identities for Negative Angles
Understanding how sine, cosine, and tangent behave with negative angles is essential. Specifically, sin(-ฮธ) = -sin(ฮธ), cos(-ฮธ) = cos(ฮธ), and tan(-ฮธ) = -tan(ฮธ). These identities help simplify expressions involving negative angles by relating them back to positive angle values.
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Fundamental Trigonometric Identities
Periodicity of Trigonometric Functions
Sine, cosine, and tangent functions repeat their values in regular intervals called periods. For sine and cosine, the period is 2ฯ, meaning sin(ฮธ + 2ฯ) = sin(ฮธ) and cos(ฮธ + 2ฯ) = cos(ฮธ). For tangent, the period is ฯ, so tan(ฮธ + ฯ) = tan(ฮธ). This property allows simplification of angles shifted by multiples of ฯ or 2ฯ.
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Expressing Trigonometric Functions in Terms of Given Variables
Given sin t = a, cos t = b, and tan t = c, the goal is to rewrite complex expressions using these variables. By applying angle identities and periodicity, each trigonometric term can be converted into a, b, or c, enabling a simplified and consistent expression in terms of the given variables.
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Related Practice
Textbook Question
In Exercises 39โ40, let ฮธ be an angle in standard position. Name the quadrant in which ฮธ lies.
tan ฮธ > 0 and sec ฮธ > 0
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Textbook Question
Find a cofunction with the same value as the given expression.
cos (3๐/8)
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Textbook Question
In Exercises 35โ40, convert each angle in radians to degrees. Round to two decimal places.
-4.8 radians
Textbook Question
In Exercises 39โ48, use a calculator to find the value of the trigonometric function to four decimal places.
sin 38ยฐ
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Textbook Question
In Exercises 31โ38, find a cofunction with the same value as the given expression. cos 2๐ 5