Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
tan θ cos θ

검증된 단계별 안내

Recall the definition of tangent in terms of sine and cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

Rewrite the given expression \(\tan \theta \cos \theta\) by substituting \(\tan \theta\) with \(\frac{\sin \theta}{\cos \theta}\), so it becomes \(\frac{\sin \theta}{\cos \theta} \times \cos \theta\).

Simplify the expression by canceling the \(\cos \theta\) in the numerator and denominator, since \(\cos \theta \neq 0\).

After cancellation, the expression simplifies to \(\sin \theta\).

Confirm that the final expression contains only sine and cosine functions of \(\theta\) and no quotients.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Definition of Tangent in Terms of Sine and Cosine

Tangent of an angle θ is defined as the ratio of sine to cosine, i.e., tan θ = sin θ / cos θ. This fundamental identity allows expressions involving tangent to be rewritten using only sine and cosine functions.

Multiplication and Simplification of Trigonometric Expressions

When multiplying trigonometric functions, apply algebraic rules to combine terms. For example, multiplying tan θ by cos θ involves substituting tan θ with sin θ / cos θ and then simplifying by canceling common factors to eliminate quotients.

Expressing Functions in Terms of a Single Variable

The goal is to rewrite expressions so that only sine and cosine of θ appear, without fractions. This often involves using identities and algebraic manipulation to remove quotients and express the result as a product or sum of sine and cosine functions.

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.tan θ cos θ