Trigonometric Identities

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

Introduction

Below, each identity is summarized and linked to examples that show how to derive the identity. In general, there are three strategies for deriving the identities:

  • Use purely trigonometry and algebra.[1]
  • Use the geometry of the circle.[2]
  • Use the geometry of the complex plane[3].

It is helpful to be familiar with the definitions of the trigonometric functions, the equation for Pythagorean Theorem and the trigonometry of the unit circle.

Identities

Tangent Identity

The tangent identity shows how 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 identity expresses how 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

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.

This figure illustrates the "proof without words" of the sum of two angles identities
Figure 1: Sum of Two Angles Identities (Visual Proof)

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

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 (Unit Circle)

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

Derive Sum of Two Angles Identities (Complex Plane)

The sum of two angles identities can be derived using the properties of the complex plane and Euler's formula.

Difference of Two Angles

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.

This figure illustrates the "proof without words" of the difference of two angles identities
Figure 2: 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

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.

Derive Difference of Two Angles Identities (Complex Plane)

Derive the difference of two angle identities using the properties of complex numbers and Euler's formula in the complex plane.

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 Angle Identities (Algebra)

This example derives the double angle identities using algebra and the sum of two angles identities.

Derive Double Angles Identities (Inscribed Angle)

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

Derive Double Angles Identities (Complex Plane)

The trigonometric double angle identites can be derived using the properties of the complex plane.

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)

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.

Derive Half Angle Identities (Inscribed Angle)

This example demosntrates how to derive the trig. half angle identities using the inscribed angle theorem.

Derive Half Angle Identities (Complex Plane)

This example derives the trig. half angle identities in the complex plane.

Links

References

  1. Derive the Trigonometric Identities
    Wumbo (internal)
  2. Derive the Trigonometric Identities (Unit Circle)
    Wumbo (internal)
  3. Derive the Trigonometric Identities (Complex Plane)
    Wumbo (internal)