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Sum and Difference Identities

Sine, cosine, or tangent of two angles that are added or subtracted.

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Sum and Difference Identities

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Because we know the basic trigonometric function values for 0, 30, 45, 60, 90, and all their multiples, we can sometimes find the function values of the different sums and differences of these angles like 15 and 75 degrees by using the sum and difference identities.

  • \sin (\alpha \pm \beta)=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta
  • \cos (\alpha \pm \beta)=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta
  • \tan (\alpha \pm \beta)=\frac{\sin (\alpha \pm \beta)}{\cos (\alpha \pm \beta)}=\frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}

The best way to memorize the sin and cos rules is to follow the chant "sine cosine cosine sine, cosine cosine sine sine."  The alpha angle will start first and alternate with the beta angle, so simply remember which one is sine and which one is cosine and whether the sine in between the two terms is plus or minus.  Tangent is the hardest to memorize, but if you find yourself not being able to memorize it, simply divide the sine identity by the cosine identity for the given angle.

Practice can be found here.

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