Solve a trigonometric equation

1
EXAMPLE 1
Solve a trigonometric equation
SOLUTION
First isolate sin x on one side of the equation.
Write original equation.
2 sin x
sin x
Divide each side by 2.
x
The other solution in the interval is
x
2
EXAMPLE 1
Solve a trigonometric equation
Moreover, because y sin x is periodic, there
will be infinitely many solutions.
You can use the two solutions found above to
write the general solution
(where n is any integer)
x
or
x
CHECK
3
EXAMPLE 1
Solve a trigonometric equation
4
EXAMPLE 2
Solve a trigonometric equation in an interval
Solve 9 tan2 x 2 3 in the interval 0 x lt2p.
9 tan2 x 2
Write original equation.
9 tan2 x
Subtract 2 from each side.
tan2 x
Divide each side by 9.
tan x
Take square roots of each side.
Using a calculator, you find that
Therefore, the general solution of the equation
is
x
x
(where n is any integer)
5
EXAMPLE 2
Solve a trigonometric equation in an interval
6
EXAMPLE 3
Solve a real-life trigonometric equation
where d is measured in feet and t is the time in
hours. If t 0 represents midnight, at what
time(s) is the water depth 7 feet?
7
EXAMPLE 3
Solve a real-life trigonometric equation
SOLUTION
Substitute 7 for d in the model and solve for t.
Substitute 7 for d.
Subtract 35 from each side.
Divide each side by 28.
cos q 1 when q 2np.
Solve for t.
8
EXAMPLE 3
Solve a real-life trigonometric equation
9
for Examples 1, 2, and 3
GUIDED PRACTICE
1. Find the general solution of the equation
2sin x 4 5.
2. Solve the equation 3csc2 x 4 in the
interval 0 x lt2p.
3. OCEANOGRAPHY In Example 3, at what time(s)
is the water depth 63 feet?
ANSWER
612 A.M and 636 P.M.