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

Verified step by step guidance

1. 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.
2. Recall the sine difference identity: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \). Here, let \( A = \frac{3\pi}{4} \) and \( B = x \).
3. Apply the identity: \( \sin\left(\frac{3\pi}{4} - x\right) = \sin\frac{3\pi}{4} \cdot \cos x - \cos\frac{3\pi}{4} \cdot \sin x \).
4. 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.
5. Substitute these values back into the expression to write \( \sin\left(\frac{3\pi}{4} - x\right) \) entirely in terms of \( \sin x \) and \( \cos x \).

Key Concepts

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.