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

Verified step by step guidance

1. Recall that \( \cos(-t) = \cos t \) because cosine is an even function.
2. Substitute \( \cos t = b \) into the expression.
3. The expression becomes \( 3b - b \).
4. Simplify the expression by combining like terms.
5. The simplified expression is \( 2b \).

Key Concepts

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent relate the angles of a triangle to the ratios of its sides. In this context, sin t = a, cos t = b, and tan t = c represent these functions evaluated at angle t. Understanding these functions is essential for manipulating and rewriting expressions in trigonometric terms.

Even and Odd Functions

Trigonometric functions exhibit properties of evenness and oddness. Specifically, cosine is an even function, meaning cos(-t) = cos(t), while sine and tangent are odd functions, implying sin(-t) = -sin(t) and tan(-t) = -tan(t). Recognizing these properties is crucial for simplifying expressions involving negative angles.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using known identities and properties. In this problem, you will need to apply the definitions of the trigonometric functions and their properties to rewrite the expression 3 cos(-t) - cos t in terms of a, b, and c. Mastery of algebraic techniques is vital for effective problem-solving in trigonometry.