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Analytic Trigonometry: Double-Angle and Half-Angle Formulas

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Analytic Trigonometry

Double-Angle and Half-Angle Formulas

Analytic trigonometry provides powerful tools for simplifying expressions, solving equations, and establishing identities using double-angle and half-angle formulas. These formulas are essential for understanding advanced trigonometric relationships and their applications in mathematics and physics.

Double-Angle Formulas

The double-angle formulas express trigonometric functions of twice an angle in terms of functions of the original angle. They are useful for finding exact values and simplifying expressions.

  • Double-Angle Formulas:

Formula

Expression

sin(2\theta)

cos(2\theta)

cos(2\theta)

cos(2\theta)

Double-angle formulas box

  • Example: If and , find and .

  • First, find using the Pythagorean identity: .

Triangle for sin and cos values

  • (since is in quadrant II, ).

Double-Angle Formula for Tangent

The double-angle formula for tangent allows you to express in terms of .

Formula

Expression

tan(2\theta)

Double-angle formula for tangent

Power-Reducing Formulas

Power-reducing formulas express squares of trigonometric functions in terms of the cosine of double angles. These are useful for simplifying expressions involving higher powers.

Formula

Expression

sin^2\theta

cos^2\theta

tan^2\theta

Power-reducing formula for sine squaredPower-reducing formula for cosine squaredPower-reducing formula for tangent squared

Establishing Trigonometric Identities

Double-angle and power-reducing formulas are often used to establish identities and simplify expressions. For example, expressing in terms of first powers:

  • Expand and simplify using power-reducing formulas.

Solving Trigonometric Equations

Double-angle formulas can be used to solve equations such as by rewriting the left side as and solving for .

Applications: Projectile Motion

Double-angle formulas are used in physics to model projectile motion. The range of a projectile launched at an angle with initial velocity is given by:

  • , where is the acceleration due to gravity.

  • Maximum range occurs when , i.e., .

Projectile motion diagram

Half-Angle Formulas

Half-angle formulas express trigonometric functions of half an angle in terms of the original angle. They are useful for finding exact values and simplifying expressions.

Formula

Expression

Basic half-angle formulas

General Half-Angle Formulas

The sign of the result depends on the quadrant in which lies.

Formula

Expression

General half-angle formulas

  • The or sign is determined by the quadrant of .

Alternative Half-Angle Formula for Tangent

There are alternative forms for the half-angle formula for tangent:

Formula

Expression

Alternative half-angle formula for tangent

Examples Using Half-Angle Formulas

  • Example: Find using the half-angle formula.

  • Let , so .

  • Use the formula: (since is in quadrant I).

  • Example: If and , find , , .

  • Determine the sign based on the quadrant for .

Summary Table: Double-Angle and Half-Angle Formulas

Type

Formula

Expression

Double-Angle

Double-Angle

Double-Angle

Half-Angle

Half-Angle

Half-Angle

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