20.4 Solving Trigonometric Equations 651
8. Let cos and arctan . Find the following, where and are positive constants. Your answers should be exact and as simple as possible.
(a) (b)
In Problems 9 through 11, simplify the expressions given that 0, .
9. (a) sin1sin
10. (a) sin1sin
11. (a) tan1tan
(b) cos1cos
(b) cos1cos
(b) tan1tan
In Problems 12 through 14, simplify the expressions given that 2 , 2.
12. (a) arcsinsin
13. (a) arcsinsin
14. (a) arctantan
(b) arccoscos
(b) arccoscos
(b) arctantan
20.4 SOLVING TRIGONOMETRIC EQUATIONS
Atrigonometricequationisanequationinwhichthevariabletobesolvedforistheargument ofatrigonometricfunction.Ifwecangetthetrigonometricequationintotheformsin , cos , or tan , where is a constant, then we can use the inverse trigonometric functions to help solve for .
EXAMPLE 20.9 Solve for .
SOLUTION
4 cos 1 1
4 cos 1 1
4 cos 2
cos 1
One solution to this equation is cos1 2 . But there are actually two solutions for
[0, 2] and inÞnitely many solutions due to the periodicity of the cosine function. For this problem we can turn to a triangle we know and love. We know cos3 1. To have a negative cosine the angle must be in the second or third quadrant. So, the solutions are
2 2 or 4 2,
where is any integer. See Figure 20.30 on the following page for illustration.
652 CHAPTER 20 TrigonometryÑCircles and Triangles
v y = cos x P(2)
3
— 1
1 u —2 — 2
3 2 2
2 y = — 1
solutions P(4)
Figure 20.30
EXAMPLE 20.10 Solve for .
2 sin2 sin 1
SOLUTION This is a quadratic equation in sin . We can either work with the sin or we can begin the problem with the substitution sin , solve for , and then return to sin . WeÕll take this latter approach.
2 2 1 2 2 1 0
12 10
1 0 or 2 1 0
1 or
sin 1 or
So
1
sin 1
2 or 2 or 5 2,
where is an integer.
v P( )
sin x
6
u —2 2
P( 6 ) or P(— 6 ) P( 6) or P(2 — 6)
1
x 1 2
Figure 20.31
20.4 Solving Trigonometric Equations 653
EXAMPLE 20.11 Solve for .
2 cos2 sin 1
SOLUTION
EXAMPLE 20.12
SOLUTION
We begin by converting the cosines into sines so we have a quadratic in sin .
2 cos2 sin 1 21 sin2 sin 1
2 2 sin2 sin 1 0 2 sin2 sin 1 0
2 sin2 sin 1 0
This is the equation we began with in the previous example, so the solutions are the same.
Find all in the interval [2, 2] such that
tan sec tan .
CAUTION DonÕt divide both sides of the equation by tan and leave yourself with only sec 1. The equation is true if tan 0. Lobbing off the tan means youÕll miss a couple of solutions.
y = tan x
tan x sec x = tan x x tan x sec x — tan x = 0
tan x (sec x — 1) = 0
y = cos x
—2 — 2 x tan x = 0
x = —2, —, 0, , 2
or sec x —1 = 0 sec x = 1 cos x = 1
or x = —2, 0, 2
Figure 20.32
Putting this all together, we get
2, , 0, , 2.
EXAMPLE 20.13
SOLUTION
Find all [0, 2] such that cos2 0.3. Give exact answers and then give numerical approximations.
LetÕs begin by letting 2. Then we can solve cos 0.3.
We want 0 2,
so 2 0 2 2 2 and 0 4.
654 CHAPTER 20 TrigonometryÑCircles and Triangles
We can get one solution to cos 0.3 by using inverse cosine.
arccos0.31.266
From Figure 20.33 we see that there is a second point, , on the unit circle with a -coordinate of 0.3. LetÕs locate with an angle between 0 and 2.
2 arccos0.3
v
1 P(1.266)
.3 1 u
Q
Figure 20.33
We want all [0, 4] such that cos 0.3.
arccos0.3,
2 arccos0.3,
and arccos0.32,
and 2 arccos0.32.
corresponding
to point corresponding
to point
, so
1 arccos0.3,
1 arccos0.3,
1 arccos0.3,
1 arccos0.3 2 1 arccos0.3.
Numerical approximations give us
0.63 or 3.77 or 2.51 to 5.65.
A sketch of cos2 and 0.3 on [0, 2] supports our solutions (and assures us that we have found all the solutions requested).
20.4 Solving Trigonometric Equations 655
y
y = .3 3 2 x 2 2
Figure 20.34
P R O B L E M S F O R S E C T I O N 2 0 . 4
1. Find all between 0 and 2 such that (a) 4 cos2 3.
(b) 2 sin2 sin 1 0. (Hint: this is a quadratic in sin .)
(c) sin 23. (Give exact answers, as well as numerical approximations.)
2. Find all solutions. (a) sin2 0.25.
(b) cos2 2 cos 1 0. (Hint: Let cos and Þrst Þnd .) (c) cos2 4 cos 3 0.
3. Find all solutions to the following equations: (a) sec2 2.
(b) cos2 0.2cos.WhycanÕtyoucancelthecos frombothsidesoftheequation? (c) sin2 3 cos 1.
4. Below is the graph of cos2 on [, ]. Find the exact coordinates of the -intercepts.
y
x
For each of the equations in Problems 5 through 13, Þnd all solutions in the interval [0, 2]. Give exact answers (as opposed to numerical approximations).
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