Summary of Trigonometric Identities

Trigonometric Identities Summary

Reciprocal Identities

• sin \theta = \frac{1}{csc \theta}

• cos \theta = \frac{1}{sec \theta}

• tan \theta = \frac{1}{cot \theta}

• csc \theta = \frac{1}{sin \theta}

• sec \theta = \frac{1}{cos \theta}

• cot \theta = \frac{1}{tan \theta}

Quotient Identities

• tan \theta = \frac{sin \theta}{cos \theta}

• cot \theta = \frac{cos \theta}{sin \theta}

Pythagorean Identities

• 1 = sin^2 \theta + cos^2 \theta

• sec^2 \theta = tan^2 \theta + 1

• csc^2 \theta = 1 + cot^2 \theta

Even/Odd Identities

• sin(- \theta) = -sin \theta

• csc(- \theta) = -csc \theta

• cos(- \theta) = cos \theta

• sec(- \theta) = sec \theta

• tan(- \theta) = -tan \theta

• cot(- \theta) = -cot \theta

Cofunction Identities

• sin(\frac{\pi}{2} - \theta) = cos \theta

• cos(\frac{\pi}{2} - \theta) = sin \theta

• tan(\frac{\pi}{2} - \theta) = cot \theta

• csc(\frac{\pi}{2} - \theta) = sec \theta

• sec(\frac{\pi}{2} - \theta) = csc \theta

• cot(\frac{\pi}{2} - \theta) = tan \theta

Sum and Difference Formulas

• sin(\alpha + \beta) = sin \alpha cos \beta + sin \beta cos \alpha

• sin(\alpha - \beta) = sin \alpha cos \beta - sin \beta cos \alpha

• cos(\alpha + \beta) = cos \alpha cos \beta - sin \alpha sin \beta

• cos(\alpha - \beta) = cos \alpha cos \beta + sin \alpha sin \beta

• tan(\alpha + \beta) = \frac{tan \alpha + tan \beta}{1 - tan \alpha tan \beta}

• tan(\alpha - \beta) = \frac{tan \alpha - tan \beta}{1 + tan \alpha tan \beta}

Double Angle Identities

• sin 2\theta = 2 sin \theta cos \theta

• tan 2\theta = \frac{2 tan \theta}{1 - tan^2 \theta}

• cos 2\theta = cos^2 \theta - sin^2 \theta = 2 cos^2 \theta - 1 = 1 - 2 sin^2 \theta

Power Reducing Identities

• sin^2 \theta = \frac{1 - cos 2\theta}{2}

• cos^2 \theta = \frac{1 + cos 2\theta}{2}

• tan^2 \theta = \frac{1 - cos 2\theta}{1 + cos 2\theta}

Half Angle Identities

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

• cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + cos \theta}{2}}

• tan \frac{\theta}{2} = \frac{1 - cos \theta}{sin \theta} = \frac{sin \theta}{1 + cos \theta}

Product to Sum Identities

• sin \alpha sin \beta = \frac{1}{2} [cos(\alpha - \beta) - cos(\alpha + \beta)]

• cos \alpha cos \beta = \frac{1}{2} [cos(\alpha - \beta) + cos(\alpha + \beta)]

• sin \alpha cos \beta = \frac{1}{2} [sin(\alpha - \beta) + sin(\alpha + \beta)]

Sum to Product Identities

• sin \alpha + sin \beta = 2 sin(\frac{\alpha + \beta}{2}) cos(\frac{\alpha - \beta}{2})

• sin \alpha - sin \beta = 2 cos(\frac{\alpha + \beta}{2}) sin(\frac{\alpha - \beta}{2})

• cos \alpha + cos \beta = 2 cos(\frac{\alpha + \beta}{2}) cos(\frac{\alpha - \beta}{2})

• cos \alpha - cos \beta = -2 sin(\frac{\alpha + \beta}{2}) sin(\frac{\alpha - \beta}{2})