Trigonometry/Identities

Let us take a right angled triangle with hypotenuse length 1. If we mark one of the acute angles as , then using the definition of the sine ratio, we have


As the hypotenuse is 1,


Repeating the same process using the definition of the cosine ratio, we have


Pythagorean identities

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Since this is a right triangle, we can use the Pythagorean Theorem:

 

 

 

This is the most fundamental identity in trigonometry.

 

 

 

 

From this identity, if we divide through by squared cosine, we are left with:

 

 

 

If instead we divide the original identity by squared sine, we are left with:

 

 

 

There are basically 3 main trigonometric identities. The proofs come directly from the definitions of these functions and the application of the Pythagorean theorem:

 
 
 


Angle sum-difference identities

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Cofunction identities

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Multiple angle identities

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