Domanda del libro di testo
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.
(0, -1)
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 1, Problema 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)
Guida verificata passo dopo passo1
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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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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Percorso guidato
5:32
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 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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5:33Period of Sine and Cosine Functions
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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6:04Introduction to Trigonometric Functions
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