Solve cos(x) = cos(a) and sin(x) = sin(a)

The general principle

A trigonometric equation is not solved in the same way as a standard equation: there are an infinite number of solutions, as the cos and sin functions are periodic with a period of 2π. We use two fundamental properties.

The case of the equation cos(x) = cos(a)

cos(x) = cos(a) <=> x = a + 2kπ or x = -a + 2kπ, where k is a relative integer (k ∈ Z).

Example: Solve cos(x) = cos(π/3) on [0, 2π[. We write x = π/3 + 2kπ or x = -π/3 + 2kπ. On [0, 2π[, this gives x = π/3 or x = 2π – π/3 = 5π/3.

The case of the equation sin(x) = sin(a)

sin(x) = sin(a) <=> x = a + 2kπ or x = π - a + 2kπ, where k is a relative integer.

Example: Solve sin(x) = 1/2 on [0, 2π[. We know that sin(π/6) = 1/2, so a = π/6. x = π/6 + 2kπ or x = π – π/6 + 2kπ = 5π/6 + 2kπ. On [0, 2π[: x = π/6 or x = 5π/6.

Notable values to be aware of

angle a	cos(a)	sin(a)
0	1	0
pi/6	sqrt(3)/2	1/2
π/4	√2/2	√2/2
π/3	1/2	√3/2
π/2	0	1

Common pitfall

Don’t forget the second type of solution (the ‘-a’ for cos, the ‘π-a’ for sin): overlooking this set means you’ll miss half the solutions. Another common mistake is confusing the two formulas for cos and sin.