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.
cos t + cos(t + 1000𝜋) - tan t - tan(t + 999𝜋) - sin t + 4 sin(t - 1000𝜋)

Verified step by step guidance

1. Recall the periodic properties of sine, cosine, and tangent functions: sine and cosine have a period of \(2\pi\), and tangent has a period of \(\pi\). This means \(\sin(t + 2k\pi) = \sin t\), \(\cos(t + 2k\pi) = \cos t\), and \(\tan(t + k\pi) = \tan t\) for any integer \(k\).
2. Simplify each trigonometric term with the given large multiples of \(\pi\) using the periodicity: - \(\cos(t + 1000\pi)\) can be rewritten using the fact that \(1000\pi = 500 \times 2\pi\), so \(\cos(t + 1000\pi) = \cos t\). - \(\tan(t + 999\pi)\) can be rewritten using \(999\pi = 999 \times \pi\), so \(\tan(t + 999\pi) = \tan t\). - \(\sin(t - 1000\pi)\) can be rewritten as \(\sin(t - 1000\pi) = \sin t\) because \(-1000\pi = -500 \times 2\pi\).
3. Substitute the simplified expressions back into the original expression: \(\cos t + \cos t - \tan t - \tan t - \sin t + 4 \sin t\).
4. Group like terms together: Combine the cosine terms: \(\cos t + \cos t = 2b\) (since \(\cos t = b\)). Combine the tangent terms: \(-\tan t - \tan t = -2c\) (since \(\tan t = c\)). Combine the sine terms: \(-\sin t + 4 \sin t = 3a\) (since \(\sin t = a\)).
5. Write the final simplified expression in terms of \(a\), \(b\), and \(c\): \$2b - 2c + 3a$.

Key Concepts

Periodic Properties 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(t + 2π) = sin t and cos(t + 2π) = cos t. Tangent has a period of π, so tan(t + π) = tan t. Understanding these periodicities allows simplification of expressions involving large multiples of π.

Trigonometric Function Definitions and Relationships

The functions sine (sin t), cosine (cos t), and tangent (tan t) are related by tan t = sin t / cos t. Knowing these definitions helps express complex trigonometric expressions in terms of given variables a, b, and c. This relationship is essential for rewriting and simplifying expressions involving multiple trigonometric terms.

Angle Addition and Subtraction Formulas

The angle addition and subtraction formulas express trigonometric functions of sums or differences of angles in terms of functions of individual angles. For example, cos(t + θ) = cos t cos θ - sin t sin θ. These formulas are useful for breaking down expressions like cos(t + 1000π) or sin(t - 1000π) into simpler terms involving sin t and cos t.