Trigonometric Identity Transformation Formulas

A comprehensive reference of all major trigonometric identity transformations. These formulas are essential for solving complex trigonometric equations, simplifying expressions, and performing calculus operations involving trigonometric functions.

1. Double Angle Formulas

Express trigonometric functions of 2α in terms of α

Sine Double Angle

sin(2α) = 2sin(α)cos(α)

Cosine Double Angle

cos(2α) = cos²(α) - sin²(α)

cos(2α) = 2cos²(α) - 1

cos(2α) = 1 - 2sin²(α)

cos(2α) = (cos(α) + sin(α))(cos(α) - sin(α))

Tangent Double Angle

tan(2α) = 2tan(α)/(1 - tan²(α))

Note: Requires tan(α) ≠ ±1 (denominator ≠ 0)

2. Half Angle Formulas

Express trigonometric functions of α/2 in terms of α

Foundation Identities

1 + cos(α) = 2cos²(α/2)

1 - cos(α) = 2sin²(α/2)

Power Reduction Formulas

sin²(α) = (1 - cos(2α))/2

cos²(α) = (1 + cos(2α))/2

tan²(α) = (1 - cos(2α))/(1 + cos(2α))

Half Angle Functions

sin(α/2) = ±√[(1 - cos(α))/2]

cos(α/2) = ±√[(1 + cos(α))/2]

tan(α/2) = ±√[(1 - cos(α))/(1 + cos(α))]

Sign determined by the quadrant of α/2

Alternative Tangent Half Angle Forms

tan(α/2) = (1 - cos(α))/sin(α)

tan(α/2) = sin(α)/(1 + cos(α))

These forms avoid the ± ambiguity

3. Sum-to-Product Formulas

Convert sums and differences of trigonometric functions to products

Sine Sum and Difference

sin(A) + sin(B) = 2sin((A+B)/2)cos((A-B)/2)

sin(A) - sin(B) = 2cos((A+B)/2)sin((A-B)/2)

Cosine Sum and Difference

cos(A) + cos(B) = 2cos((A+B)/2)cos((A-B)/2)

cos(A) - cos(B) = -2sin((A+B)/2)sin((A-B)/2)

Memory Aid

Let A = (A+B)/2 + (A-B)/2 and B = (A+B)/2 - (A-B)/2, then substitute α = (A+B)/2 and β = (A-B)/2

4. Product-to-Sum Formulas

Convert products of trigonometric functions to sums

sin(A)cos(B) = ½[sin(A+B) + sin(A-B)]

cos(A)cos(B) = ½[cos(A+B) + cos(A-B)]

sin(A)sin(B) = ½[cos(A-B) - cos(A+B)]

Derivation Note

These formulas are derived from the sine and cosine sum/difference formulas by adding or subtracting appropriate pairs.

5. Triple Angle Formulas

Express trigonometric functions of 3α in terms of α

Sine Triple Angle

sin(3α) = 3sin(α) - 4sin³(α)

sin(3α) = 4sin(α)sin(π/3 - α)sin(π/3 + α)

Cosine Triple Angle

cos(3α) = 4cos³(α) - 3cos(α)

cos(3α) = 4cos(α)cos(π/3 - α)cos(π/3 + α)

Tangent Triple Angle

tan(3α) = (3tan(α) - tan³(α))/(1 - 3tan²(α))

tan(3α) = tan(α)tan(π/3 - α)tan(π/3 + α)

Note: Requires 1 - 3tan²(α) ≠ 0

6. Special Cases and Applications

Common special values and when to use each formula type

Common Special Values

sin(15°) = (√6 - √2)/4

cos(15°) = (√6 + √2)/4

tan(15°) = 2 - √3

sin(22.5°) = √(2 - √2)/2

cos(22.5°) = √(2 + √2)/2

tan(22.5°) = √2 - 1

Formula Selection Guide

Use Double Angle When:

• Seeing expressions with 2α
• Simplifying sin(2θ), cos(2θ), tan(2θ)
• Converting between sin²θ and cos(2θ)

Use Half Angle When:

• Seeing expressions with θ/2
• Need to reduce powers (sin²θ, cos²θ)
• Working with angles like 15°, 22.5°

Use Sum-to-Product When:

• Simplifying sin A ± sin B
• Factoring trigonometric expressions
• Solving equations with sums

Use Product-to-Sum When:

• Integrating products of trig functions
• Simplifying sin A cos B expressions
• Converting products to linear combinations