Using the Addition Formulas


Use the addition formulas to derive the identities in Exercises 31–36.


sin (x − π/2) = −cos x

Verified step by step guidance

1. Start by recalling the addition formula for sine: \( \sin(a - b) = \sin a \cos b - \cos a \sin b \).
2. Apply the formula to \( \sin(x - \frac{\pi}{2}) \) by setting \( a = x \) and \( b = \frac{\pi}{2} \).
3. Substitute into the formula: \( \sin(x - \frac{\pi}{2}) = \sin x \cos \frac{\pi}{2} - \cos x \sin \frac{\pi}{2} \).
4. Evaluate the trigonometric values: \( \cos \frac{\pi}{2} = 0 \) and \( \sin \frac{\pi}{2} = 1 \).
5. Simplify the expression: \( \sin(x - \frac{\pi}{2}) = \sin x \cdot 0 - \cos x \cdot 1 = -\cos x \).

Key Concepts

Addition Formulas

Addition formulas are trigonometric identities that express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles. For example, the sine addition formula states that sin(a ± b) = sin(a)cos(b) ± cos(a)sin(b). These formulas are essential for simplifying expressions and deriving new identities in trigonometry.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are used to simplify expressions and solve equations. Common identities include the Pythagorean identities, reciprocal identities, and co-function identities, which help in transforming and manipulating trigonometric expressions.

Co-function Identities

Co-function identities relate the trigonometric functions of complementary angles. For instance, sin(π/2 - x) = cos(x) and cos(π/2 - x) = sin(x). These identities are particularly useful when working with angles that involve π/2, as they allow for the conversion between sine and cosine functions, facilitating the derivation of other identities.