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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