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5.1 The Elementary Trigonometric Identities

The Reciprocal identities

Each of the six trigonometric functions is a reciprocal of on of the other five.

\(\csc A = \dfrac{1}{\sin A}\)

\(\sin A = \dfrac{1}{\csc A}\)

\(\sec A = \dfrac{1}{\cos A}\)

\(\cos A = \dfrac{1}{\sec A}\)

\(\cot A = \dfrac{1}{\tan A}\)

\(\tan A = \dfrac{1}{\cot A}\)

The Sine/Cosine identities

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

\(\tan A = \dfrac{\sin A}{\cos A}\)

\(\cot A = \dfrac{\cos A}{\sin A}\)

\(\csc A = \dfrac{1}{\sin A}\)

\(\sec A = \dfrac{1}{\cos A}\)

The Pythagorean identities

Recall that the intersection point of the terminal side of angle \(A\) and the unit circle has coordinates \(( \cos A, \sin A )\). Thus \(\cos^2 A + \sin^2 A = 1\).

Dividing both sides of this equation by \(\cos^2 A\) and applying the sine/cosine identities yields \(1 + \tan^2 A = \sec^2 A\).

If the identity \(\cos^2 A + \sin^2 A = 1\) is divided by \(\sin^2 A\) and the sine/cosine identities applied to the result, one obtains \(\cot^2 A + 1 = \csc^2 A\)

These three identities are called the Pythagorean Identities and may appear in other forms where one of the terms is added or subtracted to the other side of the equation.

Examples of alternate forms of the identities:

\(\sin^2A=1-\cos^2A\)

\(\cos^2A=1-\sin^2A\)

\(\tan^2A=\sec^2A-1\)

\(\sec^2A-\tan^2A=1\)

\(\cot^2A=\csc^2A-1\)

\(\csc^2A-\cot^2A=1\)

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 angle \(A\) is always \(\frac{\pi}{2}-A\). Thus

\(\cos A = \sin \left( \frac{\pi}{2} – A \right)\)

\(\sin A = \cos \left( \frac{\pi}{2} – A \right)\)

\(\cot A = \tan \left( \frac{\pi}{2} – A \right)\)

\(\tan A = \cot \left( \frac{\pi}{2} – A \right)\)

\(\csc A = \sec \left( \frac{\pi}{2} – A \right)\)

\(\sec A = \csc \left( \frac{\pi}{2} – A \right)\)

The Even/Odd Identities

Recall that a function \(f\) is even if for every number \(x\) in its domain, \(-x\) is also in its domain and \(f(-x)=f(x)\).

Recall also that a function \(f\) is odd if for every number \(x\) in its domain, \(-x\) is also in its domain and \(f(-x)=-f(x)\).

Of the six trigonometric functions, only cosine and secant are even. The other four are odd.

\(\sin(-x)=-\sin(x)\)

\(\cos(-x)=\cos(x)\)

\(\tan(-x)=-\tan(x)\)

\(\csc(-x)=-\csc(x)\)

\(\sec(-x)=\sec(x)\)

\(\cot(-x)=-\cot(x)\)