Trigonometric Identities

The trigonometric identites are a set of equations derived from the properties of the right triangle. The table below shows the equation for each identity and the rest of the page explains how to derive each identity. One should be familiar with Pythagorean's Theorem and the three main trigonometric functions before studying the identities.

Name Equation
Pythagorean's Identity
Tangent
Cosecant
Secant
Cotangent
Sum of Two Angles (Sine)
Sum of Two Angles (Cosine)
Difference of Two Angles (Sine)
Difference of Two Angles (Cosine)

Pythagorean Identity

The pythagorean identity relates the sides of the right triangle together using only the angle of the right triangle. The identity is derived using pythagorean’s theorem and the properties of the unit circle. Geometrically, the components of the equation can be visualized by the right triangle below.

Pythagorean Identity

Lets start with the triangle formed by the unit circle, which visualizes all right triangles of hypotenuse one. Note, the adjacent side corresponds to the x-component of the right triangle and the opposite side corresponds to the y-component of the right triangle. The two components form the (x,y) point along the circumference of the circle.

Unit Circle Notation

Then, using the definitions of the trigonometric functions cosine and sine, we can substitute the variables which represent the adjacent side (x), the opposite side (y), and hypotenuse (1) of the right triangle into the equations.

After simplifying the equations, the adjacent side corresponds directly with the cosine function and the opposite side corresponds with the sine function for a given angle. Next, let’s recall the equation for Pythagorean's Theorem, which relates the squares of the sides together.

Pythagorean Theorem

After substituting the corresponding variables to convert the theorem into the Cartesian Coordinate System we are left with a familiar equation, the equation of a circle.

Then, by substituting the corresponding sine and cosine function above, which we found to correspond to the x and y components of the triangle, we get Pythagorean’s identity.

Trigonometric Functions

The trigonometric functions are closely related together and it can be useful to transform a function into a different shape.

Tangent

For example, the tangent function can be composed by the cosine and sine functions. This can be seen from studying the unit circle and using the same relations defined when deriving the Pythagorean Identity above.

Unit Circle Notation

If we start by writing the definition of the tangent function, then we can substitue in the cosine and sine function for and .

Secant

The secant function is the reciprocal of the cosine function. This is best visualized by the graph of the two functions. When the cosine function approaches zero, the secant function diverges to positive or negative infinity. When the cosine function approaches the value one, the secant function converges to one.

Secant Graph

Cosecant

The secant function is the reciprocal of the sine function. The graph of the two functions together shows their relationship. When the sine function approaches zero, the cosecant function diverges to positive or negative infinity. When the sine function approaches the value one, the cosecant function converges to one.

Cosecant Graph

Cotangent

The cotangent function is the reciprocal of the tangent function. The graph of the two functions together shows their relationship. When the tangent function approaches zero, the cotangent function diverges to positive or negative infinity.

Cotangent Graph

Sum of two Angles Identities

Sum of Two Angles

Difference of Two Angles Identities

Difference of Two Angles