Trigonometric Identities

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A trigonometric identity is an equation involving trigonometric functions which is true for all values of the involved variables. These identities are useful whenever expressions involving trigonometric functions have to be simplified. An important application is the integration of non-trigonometric functions: a common trick involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.

Notation: With trigonometric functions, we use the abbreviation sin2(x) for (sin(x))2.

From the Definitions

tan(x) = sin(x) / cos(x)
cot(x) = 1 / tan(x)
sec(x) = 1 / cos(x)
csc(x) = 1 / sin(x)

Periodicity, Symmetry and Shifts

These are easiest proven from the unit circle:

sin(x) = sin(x + 2π)
cos(x) = cos(x + 2π)
tan(x) = tan(x + π)
sin(- x) = - sin(x)
cos(- x) = cos(x)
tan(- x) = - tan(x)
cot(- x) = - cot(x)
sin(x) = cos(π/2 - x)
cos(x) = sin(π/2 - x)
tan(x) = cot(π/2 - x)

From the Pythagorean Theorem

sin2(x) + cos2(x) = 1
tan2(x) + 1 = sec2(x)
cot2(x) + 1 = csc2(x)

Addition Theorems

The quickest way to prove these is Euler's formula. The tangent formula follows from the other two.

sin(x +/- y) = sin(x) cos(y) +/- cos(x) sin(y)
cos(x +/- y) = cos(x) cos(y) -/+ sin(x) sin(y)
tan(x +/- y) = (tan(x) +/- tan(y)) / (1 -/+ tan(x) tan(y))

Double Angle Formulas

These can be shown by substituting x = y in the addition theorems, and using the Pythagorean formula for the latter two. Or use de Moivres formula with n = 2.

sin(2x) = 2 sin (x) cos(x)
cos(2x) = cos2(x) - sin2(x)
cos(2x) = 2 cos2(x) - 1
cos(2x) = 1 - 2 sin2(x)

Power Reduction Formulas

Solve the third and fourth double angle formula for cos2(x) and sin2.(x).

cos2(x) = (1 + cos(2x)) / 2
sin2(x) = (1 - cos(2x)) / 2

Half Angle Formulas

Substitute x/2 for x in the power reduction formulas, then solve for cos(x/2) and sin(x/2).

cos(x/2) = √((1 + cos(x)) / 2)
sin(x/2) = √((1 - cos(x)) / 2)

Products to Sums

These can be proven by expanding their right-hand-sides using the addition theorems.

cos(x) cos(y) = (cos(x+y) + cos(x-y)) / 2
sin(x) sin(y) = (cos(x-y) - cos(x+y)) / 2
sin(x) cos(y) = (sin(x+y) + sin(x-y)) / 2

Sums to Products

Replace x by (x+y) / 2 and y by (x-y) / 2 in the Product-to-Sum formulas.

sin(x) + sin(y) = 2 sin((x+y) / 2) cos((x-y) / 2)
cos(x) + cos(y) = 2 cos((x+y) / 2) cos((x-y) / 2)

Calculus

If the trigonometric functions are defined in terms of geometry, then their derivatives can be found by first verifying that limx->∞ sin x / x = 1 and then using the limit definition of the derivative and the addition theorems; if they are defined by their Taylor series, then the derivatives can be found by differentiating the power series term by term.

d/dx sin(x) = cos(x)

The rest of the trig functions can be differentiated using the above identities and the rules of differentiation, for instance

d/dx cos(x) = -sin(x)
d/dx tan(x) = sec2(x)

The integral identities can be found in Wikipedia's table of integrals.


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