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

Verified step by step guidance

1. Recall the angle addition formula for sine: \(\sin(a + b) = \sin a \cos b + \cos a \sin b\).
2. Apply the formula to \(\sin(\pi + x)\) by letting \(a = \pi\) and \(b = x\), so \(\sin(\pi + x) = \sin \pi \cos x + \cos \pi \sin x\).
3. Use the known exact values: \(\sin \pi = 0\) and \(\cos \pi = -1\).
4. Substitute these values back into the expression: \(\sin(\pi + x) = 0 \cdot \cos x + (-1) \cdot \sin x\).
5. Simplify the expression to get \(\sin(\pi + x) = -\sin x\), which expresses the function in terms of \(x\) alone.

Key Concepts

Angle Addition Formulas

Angle addition formulas express trigonometric functions of sums or differences of angles in terms of functions of individual angles. For sine, the formula is sin(a + b) = sin(a)cos(b) + cos(a)sin(b). This allows rewriting expressions like sin(π + x) using known values of sine and cosine at π.

Trigonometric Values at Special Angles

Certain angles, such as π (180°), have well-known sine and cosine values: sin(π) = 0 and cos(π) = -1. These values simplify expressions involving these angles, enabling the reduction of complex expressions like sin(π + x) to simpler forms involving sin(x) and cos(x).

Function Transformation and Periodicity

Trigonometric functions exhibit periodicity and symmetry properties, such as sin(θ + 2π) = sin(θ) and sin(π + x) = -sin(x). Understanding these transformations helps rewrite functions involving shifted angles into equivalent expressions involving the original variable alone.