Geometry Spot/Formulas/Identity Transformations

Trigonometric Identity Transformations

Essential formulas for solving complex trigonometric equations, simplifying expressions, and performing calculus operations.

Complete Reference

All Identities

1. Double Angle Formulas

Express trigonometric functions of 2α in terms of α

\\sin(2\\alpha) = 2\\sin\\alpha\\cos\\alpha

\\cos(2\\alpha) = \\cos^2\\alpha - \\sin^2\\alpha

\\cos(2\\alpha) = 2\\cos^2\\alpha - 1

\\cos(2\\alpha) = 1 - 2\\sin^2\\alpha

\\tan(2\\alpha) = \\frac{2\\tan\\alpha}{1 - \\tan^2\\alpha}

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

2. Half Angle Formulas

Express trigonometric functions of α/2 in terms of α

1 + \\cos\\alpha = 2\\cos^2\\frac{\\alpha}{2}

1 - \\cos\\alpha = 2\\sin^2\\frac{\\alpha}{2}

\\sin^2\\alpha = \\frac{1 - \\cos(2\\alpha)}{2}

\\cos^2\\alpha = \\frac{1 + \\cos(2\\alpha)}{2}

\\tan^2\\alpha = \\frac{1 - \\cos(2\\alpha)}{1 + \\cos(2\\alpha)}

\\sin\\frac{\\alpha}{2} = \\pm\\sqrt{\\frac{1 - \\cos\\alpha}{2}}

\\cos\\frac{\\alpha}{2} = \\pm\\sqrt{\\frac{1 + \\cos\\alpha}{2}}

\\tan\\frac{\\alpha}{2} = \\pm\\sqrt{\\frac{1 - \\cos\\alpha}{1 + \\cos\\alpha}}

Sign determined by the quadrant of α/2

\\tan\\frac{\\alpha}{2} = \\frac{1 - \\cos\\alpha}{\\sin\\alpha}

\\tan\\frac{\\alpha}{2} = \\frac{\\sin\\alpha}{1 + \\cos\\alpha}

These forms avoid the ± ambiguity

3. Sum-to-Product Formulas

Convert sums and differences of trigonometric functions to products

\\sin A + \\sin B = 2\\sin\\frac{A+B}{2}\\cos\\frac{A-B}{2}

\\sin A - \\sin B = 2\\cos\\frac{A+B}{2}\\sin\\frac{A-B}{2}

\\cos A + \\cos B = 2\\cos\\frac{A+B}{2}\\cos\\frac{A-B}{2}

\\cos A - \\cos B = -2\\sin\\frac{A+B}{2}\\sin\\frac{A-B}{2}

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 = \\frac{1}{2}[\\sin(A+B) + \\sin(A-B)]

\\cos A \\cos B = \\frac{1}{2}[\\cos(A+B) + \\cos(A-B)]

\\sin A \\sin B = \\frac{1}{2}[\\cos(A-B) - \\cos(A+B)]

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

\\sin(3\\alpha) = 3\\sin\\alpha - 4\\sin^3\\alpha

\\cos(3\\alpha) = 4\\cos^3\\alpha - 3\\cos\\alpha

\\tan(3\\alpha) = \\frac{3\\tan\\alpha - \\tan^3\\alpha}{1 - 3\\tan^2\\alpha}

Note: Requires 1 - 3tan²(α) ≠ 0

6. Special Cases and Applications

Common special values and when to use each formula type

\\sin 15^\\circ = \\frac{\\sqrt{6} - \\sqrt{2}}{4}

\\cos 15^\\circ = \\frac{\\sqrt{6} + \\sqrt{2}}{4}

\\tan 15^\\circ = 2 - \\sqrt{3}

\\sin 22.5^\\circ = \\frac{\\sqrt{2 - \\sqrt{2}}}{2}

\\cos 22.5^\\circ = \\frac{\\sqrt{2 + \\sqrt{2}}}{2}

\\tan 22.5^\\circ = \\sqrt{2} - 1