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 Skip Navigation LinksMath 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.

Internet references

Related Pages in this website

Trig identities

 

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