5.1 The Elementary Identities

 

The elementary identities can be classified into several categories

 

The Sine/Cosine identities.

 

Each of the trigonometric functions can be expressed in terms of sine and cosine.

 

tan A = sin A / cos A

 

cot A = cos A / sin A

 

csc A = 1 / sin A

 

sec A = 1 / cos A

 

The Pythagorean identities

 

Recall that the intersection point of the terminal side of A and the unit circle has coordinates ( cos A, sin A ).  Thus

 

cos2 A + sin2 A = 1      Dividing through by cos2 A and applying the sine/cosine identities yields

 

1 + tan2 A  = sec2 A     If the first identity is divided by sin2 A, we get

 

cot2 A + 1 = csc2 A

 

The Cofunction Identities

 

Each trigonometric function has its corresponding cofunction.  Recall that the ‘co’ in cofunction comes from ‘complementary’, as in ‘complementary angle’.  The cofunction is always the function of the complementary angle.  The complement of A is always ( π / 2 – A ).  Thus

 

cos A = sin ( π / 2 – A )

 

sin A = cos ( π / 2 – A )

 

csc A = sec ( π / 2 – A )

 

sec A = csc ( π / 2 – A )


cot A = tan ( π / 2 – A )

 

tan A = cot ( π / 2 – A )

 

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