List of trigonometric identities

These identities are useful whenever expressions involving trigonometric functions need to be simplified.

An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.

Using these identities, it is possible to express any trigonometric function in terms of any other (up to a plus or minus sign):

By examining the unit circle, one can establish the following properties of the trigonometric functions.

this is the angle determined by the free vector (starting at the origin) and the positive

satisfy simple identities: either they are equal, or have opposite signs, or employ the complementary trigonometric function.

They can also be derived by using a slightly modified version of the figure for the angle sum identities, both of which are shown here.

Terms with infinitely many sine factors would necessarily be equal to zero.

Ptolemy's theorem is important in the history of trigonometric identities, as it is how results equivalent to the sum and difference formulas for sine and cosine were first proved.

In the special cases of one of the diagonals or sides being a diameter of the circle, this theorem gives rise directly to the angle sum and difference trigonometric identities.

[17] The relationship follows most easily when the circle is constructed to have a diameter of length one, as shown here.

[21] The Chebyshev method is a recursive algorithm for finding the nth multiple angle formula knowing the

Serving a purpose similar to that of the Chebyshev method, for the tangent we can write:

The fact that the triple-angle formula for sine and cosine only involves powers of a single function allows one to relate the geometric problem of a compass and straightedge construction of angle trisection to the algebraic problem of solving a cubic equation, which allows one to prove that trisection is in general impossible using the given tools.

A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x3 − 3x + d = 0, where

None of these solutions are reducible to a real algebraic expression, as they use intermediate complex numbers under the cube roots.

The product-to-sum identities[28] or prosthaphaeresis formulae can be proven by expanding their right-hand sides using the angle addition theorems.

[citation needed] The following relationship holds for the sine function

For some purposes it is important to know that any linear combination of sine waves of the same period or frequency but different phase shifts is also a sine wave with the same period or frequency, but a different phase shift.

This is useful in sinusoid data fitting, because the measured or observed data are linearly related to the a and b unknowns of the in-phase and quadrature components basis below, resulting in a simpler Jacobian, compared to that of

These two equations can be used to solve for cosine and sine in terms of the exponential function.

For example, that ei(θ+φ) = eiθ eiφ means that That the real part of the left hand side equals the real part of the right hand side is an angle addition formula for cosine.

The equality of the imaginary parts gives an angle addition formula for sine.

The transfer function of the Butterworth low pass filter can be expressed in terms of polynomial and poles.

An efficient way to compute π to a large number of digits is based on the following identity without variables, due to Machin.

This last expression can be computed directly using the formula for the cotangent of a sum of angles whose tangents are t1, ..., tn−1 and its value will be in (−1, 1).

Note that if t = p/q is rational, then the (2t, 1 − t2, 1 + t2) values in the above formulae are proportional to the Pythagorean triple (2pq, q2 − p2, q2 + p2).

Ptolemy used this proposition to compute some angles in his table of chords in Book I, chapter 11 of Almagest.

For example, the haversine formula was used to calculate the distance between two points on a sphere.

The Dirichlet kernel Dn(x) is the function occurring on both sides of the next identity:

Trigonometric functions and their reciprocals on the unit circle. All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same. For sin, cos and tan the unit-length radius forms the hypotenuse of the triangle that defines them. The reciprocal identities arise as ratios of sides in the triangles where this unit line is no longer the hypotenuse. The triangle shaded blue illustrates the identity , and the red triangle shows that .
Unit circle with a swept angle theta plotted at coordinates (a,b). As the angle is reflected in increments of one-quarter pi (45 degrees), the coordinates are transformed. For a transformation of one-quarter pi (45 degrees, or 90 – theta), the coordinates are transformed to (b,a). Another increment of the angle of reflection by one-quarter pi (90 degrees total, or 180 – theta) transforms the coordinates to (-a,b). A third increment of the angle of reflection by another one-quarter pi (135 degrees total, or 270 – theta) transforms the coordinates to (-b,-a). A final increment of one-quarter pi (180 degrees total, or 360 – theta) transforms the coordinates to (a,-b).
Transformation of coordinates ( a , b ) when shifting the reflection angle in increments of
Unit circle with a swept angle theta plotted at coordinates (a,b). As the swept angle is incremented by one-half pi (90 degrees), the coordinates are transformed to (-b,a). Another increment of one-half pi (180 degrees total) transforms the coordinates to (-a,-b). A final increment of one-half pi (270 degrees total) transforms the coordinates to (b,a).
Transformation of coordinates ( a , b ) when shifting the angle in increments of
Illustration of angle addition formulae for the sine and cosine of acute angles. Emphasized segment is of unit length.
Diagram showing the angle difference identities for and
Diagram illustrating the relation between Ptolemy's theorem and the angle sum trig identity for sine. Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin( α + β ) = sin α cos β + cos α sin β .
Visual demonstration of the double-angle formula for sine. For the above isosceles triangle with unit sides and angle , the area ⁠ 1 / 2 ⁠ × base × height is calculated in two orientations. When upright, the area is . When on its side, the same area is . Therefore,
Cosine power-reduction formula: an illustrative diagram. The red, orange and blue triangles are all similar, and the red and orange triangles are congruent. The hypotenuse of the blue triangle has length . The angle is , so the base of that triangle has length . That length is also equal to the summed lengths of and , i.e. . Therefore, . Dividing both sides by yields the power-reduction formula for cosine: . The half-angle formula for cosine can be obtained by replacing with and taking the square-root of both sides:
Sine power-reduction formula: an illustrative diagram. The shaded blue and green triangles, and the red-outlined triangle are all right-angled and similar, and all contain the angle . The hypotenuse of the red-outlined triangle has length , so its side has length . The line segment has length and sum of the lengths of and equals the length of , which is 1. Therefore, . Subtracting from both sides and dividing by 2 by two yields the power-reduction formula for sine: . The half-angle formula for sine can be obtained by replacing with and taking the square-root of both sides: Note that this figure also illustrates, in the vertical line segment , that .
Proof of the sum-and-difference-to-product cosine identity for prosthaphaeresis calculations using an isosceles triangle
Diagram illustrating sum-to-product identities for sine and cosine. The blue right-angled triangle has angle and the red right-angled triangle has angle . Both have a hypotenuse of length 1. Auxiliary angles, here called and , are constructed such that and . Therefore, and . This allows the two congruent purple-outline triangles and to be constructed, each with hypotenuse and angle at their base. The sum of the heights of the red and blue triangles is , and this is equal to twice the height of one purple triangle, i.e. . Writing and in that equation in terms of and yields a sum-to-product identity for sine: . Similarly, the sum of the widths of the red and blue triangles yields the corresponding identity for cosine.