Lesson 5.14

Half-Angle Formulas

If double-angle formulas let you go from to , half-angle formulas let you go backward — from to . They're derived directly from the cosine double-angle identity.

Introduction

The double-angle formulas express in terms of or . By solving for the single-angle function and replacing with , you get formulas for the sine and cosine of half an angle. These let you find exact values for angles like or .

Past Knowledge

Double-angle formulas (5.13), especially .

Today's Goal

Derive and apply the half-angle formulas for sin, cos, and tan.

Future Success

Half-angle formulas are critical for calculus integration techniques (power reduction).

Key Concepts

Derivation (from Double-Angle)

Start with . Solve for :

Now let (so ):

The Half-Angle Formulas

FunctionFormula

The ± Sign

The is not “both.” You choose or based on the quadrant of . If is in a quadrant where the function is positive, use ; otherwise use .

Memory Aid: Sine vs. Cosine

Both formulas look identical except for one sign inside the radical:

  • sin → (minus)
  • cos → (plus)

Worked Examples

Basic

Finding cos 22.5°

Question: Find the exact value of .

Step 1: , so use the half-angle formula with .

Step 2: is in QI, so cosine is positive ():

Final Answer:

Intermediate

Finding sin 112.5°

Question: Find the exact value of .

Step 1: , so .

Step 2: is in QII, so sine is positive:

Final Answer:

Advanced

Using the Tangent Half-Angle (No ±)

Question: Find using the tangent half-angle formula.

Step 1: Use with :

Final Answer:

Common Pitfalls

Choosing the Wrong Sign

The ± depends on the quadrant of , not of . If , then (QII), where sine is positive but cosine is negative.

Mixing Up the ± and ∓ in Sine vs Cosine

sin uses (minus inside), cos uses (plus inside). Swapping them gives the wrong value.

Real-Life Applications

Calculus — Power Reduction Integration

In calculus, integrals like are impossible to solve directly. The half-angle/power-reduction identity converts this into a simple integral. This technique is used extensively in physics for computing energy, work, and probability distributions.

Practice Quiz

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