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

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Recall the angle addition formula for sine: \(\sin(a + b) = \sin a \cos b + \cos a \sin b\).

Apply the formula to \(\sin(\pi + x)\) by letting \(a = \pi\) and \(b = x\), so \(\sin(\pi + x) = \sin \pi \cos x + \cos \pi \sin x\).

Use the known exact values: \(\sin \pi = 0\) and \(\cos \pi = -1\).

Substitute these values back into the expression: \(\sin(\pi + x) = 0 \cdot \cos x + (-1) \cdot \sin x\).

Simplify the expression to get \(\sin(\pi + x) = -\sin x\), which expresses the function in terms of \(x\) alone.

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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.

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Quadratic Formula

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).

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Common Trig Functions For 45-45-90 Triangles

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.

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