Write each function as an expression involving functions of θ or x alone. See Example 2.
sin (3π/4 - x)

검증된 단계별 안내

Recognize that the expression \( \sin\left(\frac{3\pi}{4} - x\right) \) is a sine of a difference of two angles. We can use the sine difference identity to rewrite it.

Recall the sine difference identity: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \). Here, let \( A = \frac{3\pi}{4} \) and \( B = x \).

Apply the identity: \( \sin\left(\frac{3\pi}{4} - x\right) = \sin\frac{3\pi}{4} \cdot \cos x - \cos\frac{3\pi}{4} \cdot \sin x \).

Evaluate the trigonometric functions of the constant angle \( \frac{3\pi}{4} \): \( \sin\frac{3\pi}{4} \) and \( \cos\frac{3\pi}{4} \). These are standard values on the unit circle.

Substitute these values back into the expression to write \( \sin\left(\frac{3\pi}{4} - x\right) \) entirely in terms of \( \sin x \) and \( \cos x \).

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주요 개념

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

Angle Difference Identity for Sine

The angle difference identity states that sin(A - B) = sin A cos B - cos A sin B. This formula allows you to express the sine of a difference of two angles as a combination of sines and cosines of the individual angles, which is essential for rewriting sin(3π/4 - x).

Exact Values of Trigonometric Functions at Special Angles

Certain angles like π/4, π/3, and π/6 have known exact sine and cosine values. For example, sin(3π/4) = √2/2 and cos(3π/4) = -√2/2. Knowing these values helps simplify expressions involving these angles without using a calculator.

Function Notation and Variable Isolation

Expressing a trigonometric function in terms of θ or x alone means rewriting the expression so that the variable appears only inside the trigonometric functions, not combined with constants. This often involves applying identities to separate constants and variables clearly.

Write each function as an expression involving functions of θ or x alone. See Example 2.sin (3π/4 - x)