7.9: Trigonometric Identities II

Double-angle and half-angle trigonometric identities are derived from the fundamental sum and difference formulas and serve as essential tools for simplifying expressions, solving equations, and evaluating integrals. These identities reduce the complexity of trigonometric functions by relating functions of a multiple or fractional angle to functions of a single angle. Their applications extend across mathematics, physics, and engineering, particularly in Fourier analysis, wave mechanics, and calculus.

The double-angle identities are obtained by substituting equal angles into the sum formulas. For sine and cosine, they are:

For tangent, the double-angle identity is:

These identities are especially useful for reducing higher powers of trigonometric functions and simplifying integrals.

The half-angle identities are derived from the double-angle formulas by solving for sin²�� and cos²��. They are expressed as:

The choice of sign depends on the quadrant in which the half-angle lies. These identities are crucial in evaluating trigonometric functions for nonstandard angles and in integration techniques, such as trigonometric substitution.

At a skate park, a rider approaches a curved ramp. Its changing steepness alters the rider’s acceleration, tracing a path shaped by angular motion.

As the rider moves up or down the ramp, the path’s steepness changes—with similarities to the tangent function, which compares vertical rise to horizontal run.

Tangent is defined as sine divided by cosine for any angle measured from the horizontal.

From this definition, the double-angle identity can be derived. Recall sine and cosine of twice the angle. Substituting these into the definition of tangent leads to the result: tangent of twice the angle. When the rider launches higher, and the angle doubles, this identity predicts the new slope.

For smaller angles near the base of the ramp, where the rider enters a gentle curve, the half-angle identity becomes useful.

Recalling and applying the sine and cosine half-angle identities to the tangent definition gives an expression. Simplifying it by taking its conjugate and multiplying by one plus or one minus cosine produces two equivalent forms of the tangent half-angle identity.

These identities aid in analyzing directional changes and modeling curved motion in skate park dynamics.