Analytic Trigonometry

Double-Angle and Half-Angle Formulas

This section explores the double-angle and half-angle formulas in trigonometry, which are essential tools for simplifying expressions, solving equations, and finding exact values of trigonometric functions. These formulas are widely used in both pure and applied mathematics, including physics and engineering.

Double-Angle Formulas

The double-angle formulas allow us to express trigonometric functions of in terms of functions of . These are especially useful for simplifying expressions and solving trigonometric equations.

• Sine Double-Angle Formula:

• Cosine Double-Angle Formulas:

  • Alternative forms:

• Tangent Double-Angle Formula:

Example: Finding Exact Values Using Double-Angle Formulas

Suppose and is in the second quadrant. Find and .

• Since , use the Pythagorean identity to find : Since is in the second quadrant, , so .

• Now,

Establishing Identities Using Double-Angle Formulas

Double-angle formulas can be used to rewrite trigonometric expressions and prove identities. For example, expressing in terms of :

• From , solve for :

Solving Trigonometric Equations Using Double-Angle Formulas

Double-angle identities are useful for solving equations involving trigonometric functions. For example, to solve for in :

• Find all such that :

• Divide by 2:

Applications: Projectile Motion

Double-angle formulas are applied in physics, such as in projectile motion. The range of a projectile launched at an angle with initial velocity is:

• , where is the acceleration due to gravity.

• The maximum range occurs when , i.e., or .

Half-Angle Formulas

Half-angle formulas express trigonometric functions of in terms of or . The sign depends on the quadrant in which lies.

• Sine Half-Angle Formula:

• Cosine Half-Angle Formula:

• Tangent Half-Angle Formula:

Example: Finding Exact Values Using Half-Angle Formulas

Find using the half-angle formula:

• , so

• Since ,

Summary Table: Double-Angle and Half-Angle Formulas

Additional info: The sign in the half-angle formulas depends on the quadrant in which lies. These formulas are foundational for calculus, physics, and engineering applications.