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. 3 cos(-t) - cos t

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1. Recall the even-odd properties of trigonometric functions: cosine is an even function, so \(\cos(-t) = \cos t\).
2. Substitute \(\cos(-t)\) with \(\cos t\) in the expression: \(3 \cos(-t) - \cos t\) becomes \(3 \cos t - \cos t\).
3. Combine like terms: \(3 \cos t - \cos t = (3 - 1) \cos t = 2 \cos t\).
4. Since \(\cos t = b\), rewrite the expression in terms of \(b\): \$2b$.
5. Thus, the expression \(3 \cos(-t) - \cos t\) simplifies to \$2b$.

Key Concepts

Even and Odd Properties of Trigonometric Functions

Cosine is an even function, meaning cos(-t) = cos t. This property allows simplification of expressions involving negative angles by replacing cos(-t) with cos t, which is essential for rewriting the given expression in terms of a, b, and c.

Basic Trigonometric Identities

Understanding the fundamental identities such as sin²t + cos²t = 1 and tan t = sin t / cos t helps relate the variables a, b, and c. These identities are crucial for expressing trigonometric expressions consistently in terms of the given variables.

Expression Simplification Using Given Variables

The problem requires rewriting trigonometric expressions using the variables a = sin t, b = cos t, and c = tan t. This involves substituting and simplifying expressions by replacing trigonometric functions with their corresponding variables to achieve the desired form.