Solve sine x equals a half, over a full period

Inverse sine of a half gives 30 degrees. The second solution is the supplement: 180 minus 30. Both lie inside one period. x equals 30 degrees or 150 degrees

Solve cosine x equals zero, over a full period

Inverse cosine of zero gives 90 degrees. For cosine the other solution is 360 minus the first. That gives 270 degrees. x equals 90 degrees or 270 degrees

Solve 2 sine squared x minus sine x equals zero, over a full period

Take the common factor: sine x times (2 sine x − 1) equals zero. The first factor gives sine x equals zero, so 0 and 180 degrees. The second gives sine x equals a half, so 30 and 150 degrees. 0, 30, 150 and 180 degrees

Mathematics, five units · Grade 12

How do you solve a trigonometric equation over a full period?

Find one solution with the inverse function, then use the symmetry of the function to find the second one in the period, and finally add the period to generate the rest. The number of solutions per period is fixed: two for sine and cosine, one for tangent.

Learning objectives

Write the general solution of a basic trigonometric equation
Reduce an equation containing two functions to one function by an identity
List the solutions that fall inside a given interval
Discard a solution that lies outside the domain of the original expression

One solution, then all the rest

The inverse function on a calculator returns exactly one angle, the principal value. It is a correct solution and it is almost never the only one.

The second comes from symmetry. For sine, the other solution in the period is the supplement — a hundred and eighty degrees minus the first. For cosine, it is the negative of the first, or equivalently three hundred and sixty minus it.

Then add whole periods to each. That is what generates the complete family, and the question usually restricts it to one period or to a stated range.

How many solutions per period

Sine and cosine each take every value in their range twice per full turn, so an equation in one of them has two solutions in a period. Tangent takes every value once per half turn, so it has one.

Counting before solving is a genuine check. If you found one solution to a sine equation and the question asked for a full period, you are not finished.

When the equation is not simple

If the argument is a compound expression, solve for the whole argument first and substitute back at the very end. Dividing by the coefficient too early loses solutions.

If more than one function appears, use an identity to reduce it to one. An equation mixing sine and cosine usually becomes a quadratic in a single function, which then factorises.

Worked examples

Solve sine x equals a half, over a full period
1. Inverse sine of a half gives 30 degrees
2. The second solution is the supplement: 180 minus 30
3. Both lie inside one period
Answer: x equals 30 degrees or 150 degrees
Solve cosine x equals zero, over a full period
1. Inverse cosine of zero gives 90 degrees
2. For cosine the other solution is 360 minus the first
3. That gives 270 degrees
Answer: x equals 90 degrees or 270 degrees
Solve 2 sine squared x minus sine x equals zero, over a full period
1. Take the common factor: sine x times (2 sine x − 1) equals zero
2. The first factor gives sine x equals zero, so 0 and 180 degrees
3. The second gives sine x equals a half, so 30 and 150 degrees
Answer: 0, 30, 150 and 180 degrees

Common mistakes

Stopping at the calculator's answer: The inverse function returns one angle out of the two in the period. Reporting it alone answers a different question from the one asked.
Dividing by a trigonometric function: If that function can be zero, dividing by it erases solutions. Take it out as a common factor instead, exactly as with x in a quadratic.
Dividing the argument by its coefficient too early: Solve for the whole argument first, collect every solution in its range, and only then divide. Dividing first throws away solutions that were inside the enlarged range.

What to remember

The calculator gives one solution of two.
Sine: the supplement. Cosine: 360 minus it.
Two per period for sine and cosine, one for tangent.
Solve for the whole argument before dividing.