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Ch. 1 - Angles and the Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 1 - Angles and the Trigonometric FunctionsProblem 39
Chapter 1, Problem 39

In Exercises 33–42, let sin t = a, cos t = b, and tan t = c. Write each expression in terms of a, b, and c.
sin(-t - 2𝜋) - cos(-t - 4𝜋) - tan(-t - 𝜋)

Verified step by step guidance
1
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(-θ) = -sin(θ), cos(-θ) = cos(θ), and tan(-θ) = -tan(θ). These identities help simplify expressions involving negative angles by relating them back to positive angle values.
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Periodicity of Trigonometric Functions

Sine, cosine, and tangent functions repeat their values in regular intervals called periods. For sine and cosine, the period is 2π, meaning sin(θ + 2π) = sin(θ) and cos(θ + 2π) = cos(θ). For tangent, the period is π, so tan(θ + π) = tan(θ). This property allows simplification of angles shifted by multiples of π or 2π.
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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
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In Exercises 37–38, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of θ, or state that the function is undefined.

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In Exercises 31–38, find a cofunction with the same value as the given expression. cos 2𝜋 5