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 Math Help > Trigonometry > Trig Equivalences > Solving Trig Equations

This page contains a summary of methods for solving trig equations.

In Trigonometry, and again in Calculus, you will run into numerous cases in which special conversion rules, or identities, which I call "equivalences" can be used.  To motivate you to read this page, here's an example of a problem that needs two trigonometric identities:

### Example 1: Solve cos(x/2) = -sin(x - π/2)

Solve for x:

cos(x/2) = -sin(x - π/2)

The first equivalence I need is

cos(x) = -sin(x-π/2)

This is one of a whole family of "phase shift" equivalences involving adding and subtracting a right angle (π/2).  The next one I need comes from the Law of Cosines (see the isosceles triangle near the bottom of Law of Cosines 2).  It is

cos x = 2 cos²(x/2) - 1

Using these two facts, you can solve the equation:

cos(x/2) = -sin(x - π/2)
cos(x/2) = cos(x)
cos(x) - cos(x/2) = 0
2cos²(x/2)-1 - cos(x/2) = 0
2cos²(x/2) - cos(x/2) - 1 = 0

Using the quadratic formula,

cos(x/2) = 1/4 +/- sqrt(1+8)/4
cos(x/2) = 1/4 +/- 3/4
cos(x/2) = 1 or cos(x/2) -1/2
x/2=0 or x/2=2π/3
x=0 or x=4π/3

And of course, there are infinitely many other answers, which you can get using the periodicity of the cos function.

See identities for a selection of trigonometric equivalences, with proofs or explanations where needed.

### Related Pages in this website

Trig identities

The webmaster and author of this Math Help site is Graeme McRae.