Trigonometric Identities

The trigonometric identites are a set of equations derived from the properties of the right triangle and the circle. The equations are useful for manipulating and transforming math expressions.

EquationName
Tangent Identity
Cotangent Identity
Pythagorean's Identity
Sum of Two Angles (Sine)
Sum of Two Angles (Cosine)
Difference of Two Angles (Sine)
Difference of Two Angles (Cosine)
Double-Angle Identity (Sine)
Double-Angle Identity (Cosine)
Double-Angle Identity (Cosine 2)
Double-Angle Identity (Cosine 3)
Half-Angle Identity (Cosine)
Half-Angle Identity (Sine)

Identities

Below, each identity is summarized and linked to examples that show how to derive the identity. There are three general full-path strategies for deriving the identities. One should be familiar with the definitions of the trigonometric functions, the equation for Pythagorean's Theorem and the trigonometry of the unit circle.

Tangent Identity

The tangent function can be defined by the cosine and sine functions. The simplest way to see this connection is to define the length of the adjacent and opposite sides of the right triangle in terms of the and of the angle and the hypotenuse of the triangle denoted with the variable .

Right Triangle in Terms of Sine and Cosine

Then, by substituting these lengths into the definition of tangent, the length of the hypotenuse cancels, leaving the tangent identity.

Geometrically, the tangent length of an angle is equal to the length of the line tangent to the point the angle describes on the unit circle and the point where the tangent line intersects the x-axis. This length can be solved for using sine and cosine to derive the same formula.

Tangent Geometry on The Unit Circle

The graph of tangent illustrates how the tangent length changes with respect to the angle . Note, the graph of tangent diverges to infinity where the cosine function is .

Tangent Graph

Cotangent Identity

The cotangent of an angle is the reciprocal of the tangent of an angle. On the unit circle, the cotangent of an angle is equal to the adjacent side of a right triangle with an opposite side of length . Geometrically, the cotangent the corresponding length on the tangent line to the point on the circle.

Cotangent Geometry on The Unit Circle

The graph of cotangent and tangent shows the reciprocal relationship between the two functions. As tangent approaches so does cotangent and as tangent approaches cotangent divereges to infinity.

Cotangent Graph

Pythagorean Identity

The pythagorean identity relates Pythagorean’s theorem to the geometry of a right-triangle on the unit circle. Pythagorean’s theorem equates the squared lengths of a right triangle together:

Pythagorean Theorem

The cosine and sine of an angle give the coordinates of a point along the unit circle. Substituting these lengths into the equation for pythagorean’s theorem yields Pythagorean’s identity.

Unit Circle Point Defined by Cosine and Sine

Sum of two Angles Identities

The sum of two angles identities express the cosine and sine of the sum in terms of the sine and cosine of the individual angles. The identities can be derived three ways: 1) By using the geometry of a right triangle. 2) By using the geometry of the right-triangle on the circle. 3) By using the geometry of the complex plane.

Sum of Two Angles Identities

The figure above demonstrates how the identities can be formulated purely from the geometry of the right-triangle. See all derivations below:

Derive Sum of Two Angles Identities | Example

To derive the sum of two angles identities, two right triangles are placed next to eachother so their angles sum together, then their proportions are related together.

Derive Sum of Two Angles Addition Formula | Example

The sum of two angles addition formula can be derived using a quadrilateral inscribed on a circle of diameter 1.

Difference of two Angles Identities

The difference of two angles identities express the cosine and sine of the difference in terms of the sine and cosine of the individual angles. The “proof without words” below demonstrates one way to derive the identities.

Difference of Two Angles Identities

To derive the idientities step-by-step using the geometry of the image above see the example below:

Derive Difference of Two Angles Identities | Example

To derive the difference of two angles identities, two right triangles are placed next to eachother so their angles sum together to be one angle and one triangle's angle is the difference of the sum and the other.

Double Angle Identities

The double angle identities express the cosine and sine of a double angle in terms of the sine and cosine of the single angle. The identities can be derived three ways: 1) By using the previously derived theorems on this page such as Pythagorean’s Identity and the Sum of Two Angles identities. 2) By using the geometry of the inscribed angle theorem and the formula for area of a triangle. 3) By using the complex plane and the properties of complex numbers.

Double Angle Identities

The figure above demonstrates the inscribed angle theorem and the properties of similar triangles can be used to derive the double angle identities:

Derive Double Angles Identities (Algebra) | Example

The double angle identities can be derived using the sum of two angle identities in combination with Pythagorean's identity.

Derive Double Angles Identities (Inscribed Angle) | Example

The double angle identities can be derived using the inscribed angle theorem on the circle of radius one.

Half Angle Identities

The half angle identities express the cosine and sine of a half-angle in terms of the sine and cosine of the single angle. The identities can be derived using the geometry of the inscribed angle shown below:

Double Angle Identities

The half-angle identities can also be derived using the double angle formulas and the previous theorems derived above.

Derive Half Angle Identities (Algebra) | Example

The half-angle identities can be derived from the double angle identities by transforming the angles using algebra and then solving for the half-angle expression.

References

Unit Circle Trigonometry | Concept

The unit circle demonstrates the output of the trigonometric functions sine, cosine and tangent through the shared geometry of the right-triangle defined by a point on the perimeter of the circle.

Footnotes

  1. Derive the Trigonometric Identities using the geometry of the Complex Plane