**Home**
/ **Review
Topics**

## Solving
Trigonometric Equations

A trigonometric equation
is an equation that contains trigonometric functions such as sine
and cosine. There are usually two ways to solve a trigonometric equation,
an algebraic method and a graphical method. When solving algebraically,
we first isolate the trigonometric function and then use the inverse
of that function to solve for the variable. Since the trigonometric
functions are periodic, a trigonometric equation can have infinitely
many solutions. Depending on the situation, we may be asked to solve
the equation over a specified interval or we may be asked to find
all solutions of the equation. When we are asked to find all solutions
of a trigonometric equation, we first find the solutions over the
interval representing one complete cycle of the graph, and then use
the fact that the trigonometric functions are periodic to find the
remaining solutions.

### Example 1

Find all solutions of the
equation _{}.

### Solution

The first step is to solve
for _{}.

_{}

One solution to this equation
is _{}. While the inverse sine gives only one solution,
there are actually an infinite number of solutions. Since the sine
function has period 2p, we should first find all solutions
in the interval _{}. Then we can add any integer multiple of 2p
to these solutions to get another solution. To find all solutions
in the interval _{}, we can think of t as the radian measure of an
angle in standard position whose terminal side intersects the unit
circle at a point with y-coordinate ½. There are exactly two such
angles, one in Quadrant I, and the other in Quadrant II. See Figure
5.

Figure 5

We already know that the
angle in quadrant I has radian measure _{} Using the symmetry of the circle, we can easily
determine that the angle in Quadrant II is _{} Finally, we see that all solutions
are of the form,

_{}

We
can also write the solutions as decimals,

_{}.

### Example 2

Find all solutions to the
equation _{} over the interval _{}.

### Solution

We
first solve for cos t,

_{}

The inverse cosine
function gives _{} Since the cosine function is also negative
in Quadrant III, there is another solution in that quadrant. It is
_{}(rounded to four decimal places). See Figure 6.

Figure 6