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.
|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)|
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.
- Use purely algebra and trigonometry.
- Use the geometry of the unit circle and theorems such as the double-angle theorem and Ptolemy’s theorem.
- Use the geometry of the complex plane.
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 .
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.
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 .
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.
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.
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:
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.
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.
The figure above demonstrates how the identities can be formulated purely from the geometry of the right-triangle. See all derivations below:
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.
The sum of two angles addition formula can be derived using a quadrilateral inscribed on a circle of diameter 1.
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.
To derive the idientities step-by-step using the geometry of the image above see the example below:
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.
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.
The figure above demonstrates the inscribed angle theorem and the properties of similar triangles can be used to derive the double angle identities:
The double angle identities can be derived using the sum of two angle identities in combination with Pythagorean's identity.
The double angle identities can be derived using the inscribed angle theorem on the circle of radius one.
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:
The half-angle identities can also be derived using the double angle formulas and the previous theorems derived above.
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.
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.
- Derive the Trigonometric Identities using the geometry of the Complex Plane