Summary of Trigonometric Identities

Trigonometric Identities Summary

Reciprocal Identities

  • sinθ=1cscθsin \theta = \frac{1}{csc \theta}

  • cosθ=1secθcos \theta = \frac{1}{sec \theta}

  • tanθ=1cotθtan \theta = \frac{1}{cot \theta}

  • cscθ=1sinθcsc \theta = \frac{1}{sin \theta}

  • secθ=1cosθsec \theta = \frac{1}{cos \theta}

  • cotθ=1tanθcot \theta = \frac{1}{tan \theta}

Quotient Identities

  • tanθ=sinθcosθtan \theta = \frac{sin \theta}{cos \theta}

  • cotθ=cosθsinθcot \theta = \frac{cos \theta}{sin \theta}

Pythagorean Identities

  • 1=sin2θ+cos2θ1 = sin^2 \theta + cos^2 \theta

  • sec2θ=tan2θ+1sec^2 \theta = tan^2 \theta + 1

  • csc2θ=1+cot2θcsc^2 \theta = 1 + cot^2 \theta

Even/Odd Identities

  • sin(−θ)=−sinθsin(- \theta) = -sin \theta

  • csc(−θ)=−cscθcsc(- \theta) = -csc \theta

  • cos(−θ)=cosθcos(- \theta) = cos \theta

  • sec(−θ)=secθsec(- \theta) = sec \theta

  • tan(−θ)=−tanθtan(- \theta) = -tan \theta

  • cot(−θ)=−cotθcot(- \theta) = -cot \theta

Cofunction Identities

  • sin(π2−θ)=cosθsin(\frac{\pi}{2} - \theta) = cos \theta

  • cos(π2−θ)=sinθcos(\frac{\pi}{2} - \theta) = sin \theta

  • tan(π2−θ)=cotθtan(\frac{\pi}{2} - \theta) = cot \theta

  • csc(π2−θ)=secθcsc(\frac{\pi}{2} - \theta) = sec \theta

  • sec(π2−θ)=cscθsec(\frac{\pi}{2} - \theta) = csc \theta

  • cot(π2−θ)=tanθcot(\frac{\pi}{2} - \theta) = tan \theta

Sum and Difference Formulas

  • sin(α+β)=sinαcosβ+sinβcosαsin(\alpha + \beta) = sin \alpha cos \beta + sin \beta cos \alpha

  • sin(α−β)=sinαcosβ−sinβcosαsin(\alpha - \beta) = sin \alpha cos \beta - sin \beta cos \alpha

  • cos(α+β)=cosαcosβ−sinαsinβcos(\alpha + \beta) = cos \alpha cos \beta - sin \alpha sin \beta

  • cos(α−β)=cosαcosβ+sinαsinβcos(\alpha - \beta) = cos \alpha cos \beta + sin \alpha sin \beta

  • tan(α+β)=tanα+tanβ1−tanαtanβtan(\alpha + \beta) = \frac{tan \alpha + tan \beta}{1 - tan \alpha tan \beta}

  • tan(α−β)=tanα−tanβ1+tanαtanβtan(\alpha - \beta) = \frac{tan \alpha - tan \beta}{1 + tan \alpha tan \beta}

Double Angle Identities

  • sin2θ=2sinθcosθsin 2\theta = 2 sin \theta cos \theta

  • tan2θ=2tanθ1−tan2θtan 2\theta = \frac{2 tan \theta}{1 - tan^2 \theta}

  • cos2θ=cos2θ−sin2θ=2cos2θ−1=1−2sin2θcos 2\theta = cos^2 \theta - sin^2 \theta = 2 cos^2 \theta - 1 = 1 - 2 sin^2 \theta

Power Reducing Identities

  • sin2θ=1−cos2θ2sin^2 \theta = \frac{1 - cos 2\theta}{2}

  • cos2θ=1+cos2θ2cos^2 \theta = \frac{1 + cos 2\theta}{2}

  • tan2θ=1−cos2θ1+cos2θtan^2 \theta = \frac{1 - cos 2\theta}{1 + cos 2\theta}

Half Angle Identities

  • sinθ2=±1−cosθ2sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - cos \theta}{2}}

  • cosθ2=±1+cosθ2cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + cos \theta}{2}}

  • tanθ2=1−cosθsinθ=sinθ1+cosθtan \frac{\theta}{2} = \frac{1 - cos \theta}{sin \theta} = \frac{sin \theta}{1 + cos \theta}

Product to Sum Identities

  • sinαsinβ=12[cos(α−β)−cos(α+β)]sin \alpha sin \beta = \frac{1}{2} [cos(\alpha - \beta) - cos(\alpha + \beta)]

  • cosαcosβ=12[cos(α−β)+cos(α+β)]cos \alpha cos \beta = \frac{1}{2} [cos(\alpha - \beta) + cos(\alpha + \beta)]

  • sinαcosβ=12[sin(α−β)+sin(α+β)]sin \alpha cos \beta = \frac{1}{2} [sin(\alpha - \beta) + sin(\alpha + \beta)]

Sum to Product Identities

  • sinα+sinβ=2sin(α+β2)cos(α−β2)sin \alpha + sin \beta = 2 sin(\frac{\alpha + \beta}{2}) cos(\frac{\alpha - \beta}{2})

  • sinα−sinβ=2cos(α+β2)sin(α−β2)sin \alpha - sin \beta = 2 cos(\frac{\alpha + \beta}{2}) sin(\frac{\alpha - \beta}{2})

  • cosα+cosβ=2cos(α+β2)cos(α−β2)cos \alpha + cos \beta = 2 cos(\frac{\alpha + \beta}{2}) cos(\frac{\alpha - \beta}{2})

  • cosα−cosβ=−2sin(α+β2)sin(α−β2)cos \alpha - cos \beta = -2 sin(\frac{\alpha + \beta}{2}) sin(\frac{\alpha - \beta}{2})