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Trigonometric Identities


In mathematics, an "identity" is an equation which is always true. These can be "trivially" true, like "x = x" or usefully true, such as the Pythagorean Theorem's "a2 + b2 = c2" for right triangles. There are loads of trigonometric identities, but the following are the ones you're most likely to see and use.

Basic & Pythagorean, Angle-Sum & -Difference, Double-Angle, Half-Angle, Sum, Product


Basic and Pythagorean Identities

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

Notice how a "co-(something)" trig ratio is always the reciprocal of some "non-co" ratio. You can use this fact to help you keep straight that cosecant goes with sine and secant goes with cosine.

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

The above, because they involve squaring and the number 1, are the "Pythagorean" identities. You can see this clearly if you consider the unit circle, where sin(t) = y, cos(t) = x, and the hypotenuse is 1.

sin(t) = sin(t)       cos(t) = cos(t)       tan(t) = tan(t)

Notice in particular that sine and tangent are odd functions, while cosine is an even function.

Angle-Sum and -Difference Identities

sin(α + β) = sin(α)cos(β) + cos(α)sin(β)
sin(α β) = sin(α)cos(β) cos(α)sin(β)

cos(α + β) = cos(α)cos(β) sin(α)sin(β)

cos(α β) = cos(α)cos(β) + sin(α)sin(β)

  
tan(a + b) = [tan(a) + tan(b)] / [1 - tan(a)tan(b)], tan(a - b) = [tan(a) - tan(b)] / [1 + tan(a)tan(b)] 

Double-Angle Identities

sin(2x) = 2sin(x)cos(x)

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

tan(2x) = [2 tan(x)] / [1 - tan^2(x)]

Half-Angle Identities   Copyright Elizabeth Stapel 2010-2011 All Rights Reserved

sin(x/2) = +/- sqrt[(1 - cos(x))/2], cos(x/2) = +/- sqrt[(1 + cos(x))/2], tan(x/2) = +/- sqrt[(1 - cos(x))/(1 + cos(x))]

The above identities can be re-stated as:

sin2(x) = [1 cos(2x)]

cos2(x) = [1 + cos(2x)]

tan^2(x) = [1 - cos(2x)] / [1 + cos(2x)]

Sum Identities

sin(x)+sin(y)=2sin[(x+y)/2]cos[(x-y)/2], sin(x)-sin(y)=2cos[(x+y)/2]sin[(x-y)/2], cos(x)+cos(y)=2cos[(x+y)/2]cos[(x-y)/2], cos(x)-cos(y)=-2sin[(x+y)/2]sin[(x-y)/2]

Product Identities

sin(x)cos(y)=(1/2)[sin(x+y)+sin(x-y)], cos(x)sin(y)=(1/2)[sin(x+y)-sin(x-y)], cos(x)cos(y)=(1/2)[cos(x-y)+cos(x+y)], sin(x)sin(y)=(1/2)[cos(x-y)-cos(x+y)]

You will be using all of these identities, or nearly so, for proving other trig identities and for solving trig equations. However, if you're going on to study calculus, pay particular attention to the restated sine and cosine half-angle identities, because you'll be using them a lot in integral calculus.

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Cite this article as:

Stapel, Elizabeth. "Trigonometric Identities." Purplemath. Available from
    http://www.purplemath.com/modules/idents.htm. Accessed
 

 



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