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 sim…
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.
View the full transcript and gain access to JoVE Core videos
Q1: How is the tangent function defined in terms of sine and cosine?
Tangent is defined as sine divided by cosine for any angle measured from the horizontal. This fundamental definition compares the vertical rise to the horizontal run of an angle, making it useful for analyzing slopes and directional changes in curved motion, such as a rider's path on a ramp.
Q2: What is the double-angle identity for tangent?
The double-angle identity for tangent is derived by substituting sine and cosine double-angle formulas into the tangent definition. When an angle doubles, this identity predicts the new slope or steepness, making it essential for modeling situations where angular motion increases, such as a rider launching higher on a curved ramp.
Q3: When is the half-angle identity for tangent most useful?
The half-angle identity for tangent is useful for analyzing smaller angles near the base of a ramp or gentle curves where the rider enters with reduced steepness. It simplifies expressions involving fractional angles by relating them to functions of a single angle, aiding in integration and evaluating nonstandard angles.
Q4: How are half-angle identities derived from double-angle formulas?
Half-angle identities are derived by solving the double-angle formulas for sin²(θ/2) and cos²(θ/2). The choice of sign in these identities depends on the quadrant in which the half-angle lies, allowing accurate evaluation of trigonometric functions for angles that are not standard or easily computed.
Q5: Why are double-angle and half-angle identities important in calculus?
Double-angle and half-angle identities reduce the complexity of trigonometric expressions and are crucial for simplifying integrals and solving equations. They are especially useful for reducing higher powers of trigonometric functions and enabling trigonometric substitution techniques in integration and advanced mathematical applications.
Q6: How do trigonometric identities relate to real-world applications like ramp dynamics?
Trigonometric identities model curved motion by relating changing angles to slopes and accelerations. In a skate park, the tangent function describes the ramp's steepness, while double-angle and half-angle identities predict how the slope changes as the rider moves up or down, enabling analysis of directional changes and motion dynamics.
Q7: What are the two equivalent forms of the tangent half-angle identity?
The tangent half-angle identity has two equivalent forms obtained by simplifying the ratio of sine and cosine half-angle expressions using conjugate multiplication. These forms relate tan(θ/2) to either (1 - cos θ)/sin θ or sin θ/(1 + cos θ), providing flexibility in solving different types of problems.