Trig identities

All of these are provable on their own but they are helpful to memorize because they are used all over the place in proofs and problem solving.

In this file x === theta (𝛳)

Pythagorean identities

These three relationships are all derived from the Pythagorean theorem (and easy to prove on a unit circle):

sin^2x + cos^2x = 1
1 + cot^2x = csc^2x
1 + tan^2x = sec^2x

These get used frequently like sin^2x = 1 - cos^2x

Reciprocal identities

cscx = 1/sinx
secx = 1/cosx
cotx = cosx/sinx

Quotient identities

If you define a right triangle in the unit circle (radius = 1) you can easily find the quotient identities

tanx = sinx/cosx
cotx = cosx/sinx

Co-function identities

The value of a trig function of an angle is equal to the value of the co-function of the angle's complement

const Y = pi/2

sinx = cos(Y-x)
cscx = sec(Y-x)
cosx = sin(Y-x)
secx = csc(Y-x)
tanx = cot(Y-x)
cotx = tan(Y-x)

Even-odd identities

sin(-x) = -sinx
cos...
tan...
cot...

Double angle identities

Used to transform 2(x) to x

sin(2x) != 2sin(x)

To remove the constant from the angle you need to use a double angle identity

sin2x = 2sinx * cosx
cos2x = cos^2x-sin^2x = 1-2sin^2x = 2cos^2x-1
tan2x = 2tanx/1-tan^2x

Another helpful way to write these

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

Remember to use Pythagorean identities if you are only provided with sin or cos to find the other one.

Half-angle identities

Used to transform x/2 to x

cos(x/2) = ± sqt(1+cosx/2)
sin(x/2) = ± sqt(1-cosx/2)
tan(x/2) = ± sqt(1-cosx)/sqt(1+cosx)

Use positive or negative result depending on the quadrant of the angle

Sum-difference identities

sin(x+a) = sinxcosa + cosxsina
sin(x-a) = sinxcosa - cosxsina
tan(x+y) = (tanx+tany)/(1-tanxtany)
tan(x-y) = (tanx-tany)/(1+tanxtany)

This can be useful if given an angle that is not in the set of angles presented by the unit circle

find sin(pi/12)

pi/12 = (pi/3 - pi/4)

sin(pi/12) = (sqt3/2)(sqt2/2) - (1/2)(sqt2/2)

(sqt6 - sqt2)/4

Product to sum (or difference) identities

These can be derived from the sum-difference identities

sinx * cosy = 1/2[sin(x+y)+sin(x-y)]
cosx * siny = 1/2[sin(x+y)-sin(x-y)]
cosx * cosy = 1/2[cos(x+y)+cos(x-y)]
sinx * siny = 1/2[cos(x-y)-cos(x+y)]

Sum (or difference) to product identities

sinx + siny = 2sin((x + y)/2) * cos((x - y)/2)
sinx - siny = 2cos((x + y)/2) * sin((x - y)/2)
cosx + cosy = 2cos((x + y)/2) * cos((x - y)/2)
sinx - siny = -2sin((x + y)/2) * sin((x - y)/2)

Using identities in proofs

Trig identities are frequently used in proofs by manipulating an equation until it matches an identity

Some hot tips:

  1. Stick to sin and cosine as much as possible and convert the others
  2. Make sure all angles are the same (simplify sin(2x) to sinx)
  3. Add/remove powers with the Pythagorean identities
  4. Consider if the equation is factorable
  5. Link distant functions with identities
  6. Separate constants by multiplication by the conjugate

Just practice these bad bois

Sometimes the value of 𝛳 will be constrained Ex: 𝛳 ∈ [0,2pi)