In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression. sin 22.5°

Key Concepts

Double-Angle Formulas

Double-angle formulas are trigonometric identities that express trigonometric functions of double angles in terms of single angles. For example, the sine double-angle formula states that sin(2θ) = 2sin(θ)cos(θ). These formulas are essential for simplifying expressions involving angles that are multiples of known angles.

Half-Angle Formulas

Half-angle formulas allow us to find the sine, cosine, and tangent of half of a given angle. For sine, the formula is sin(θ/2) = √((1 - cos(θ))/2). This is particularly useful for calculating the sine of angles that are not standard, such as 22.5°, which is half of 45°.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to the precise values of sine, cosine, and tangent for specific angles, often expressed in terms of square roots. For example, sin(30°) = 1/2 and cos(45°) = √2/2. Knowing these exact values helps in applying the double- and half-angle formulas effectively to find the sine of non-standard angles.