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 \(\tan(-x) = -\tan x\).
2. Use the periodicity of the functions: \(\sin(x - 2\pi) = \sin x\), \(\cos(x - 4\pi) = \cos x\), and \(\tan(x - \pi) = \tan x\).
3. 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 \(\tan(-(t + \pi)) = -\tan(t + \pi) = -\tan t = -c\).
4. Substitute the simplified terms back into the original expression: \(\sin(-t - 2\pi) - \cos(-t - 4\pi) - \tan(-t - \pi) = (-a) - b - (-c)\).
5. Simplify the expression by combining like terms: \(-a - b + c\).

Key Concepts

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.

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.

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.