Solving Trigonometric Equations: Solutions
Here are some solutions to the Exercises to accompany the section Solving
Trigonometric Equations.
Inverse Trigonometric Functions
- Use the formulas from the text for sin-1,
cos-1, and tan-1 to write formulas for all solutions
to the following equations. Note: If you recognize exact values using the table
in the text, you may give exact values for these inverse trigonometric
functions. Otherwise, you should simply use your calculator to give approximate values.
- -
= sin(x)
- Solution
- Since the table
says that
= sin(p/4), and sin is odd,
we know that -
= sin(-p/4), so sin-1(-)
= -p/4. According to formula for sin-1,
the set of all solutions to -
= sin(x)
is then x = 2kp
+ sin-1(-), (2k + 1)p - sin-1(-)
or x = 2kp
- p/4, (2k + 1)p
+ p/4. If we simply use a
calculator, we would obtain the approximate value of
0.7854 in place of p/4.
- 0.7 = sin(x)
- Solution
- According to formula for sin-1,
the set of all solutions to 0.7 = sin(x)
is x = 2kp
+ sin-1(0.7), (2k + 1)p - sin-1(0.7)
or x » 2kp
+ 0.7754, (2k + 1)p - 0.7754.
- -2 = cos(x)
- Solution
- While you might naively say that the formula for
cos-1 gives the set of all solutions to -2 = cos(x)
as x = 2kp
± cos-1(-2), cos-1(-2) is
undefined. That is, if we were to read the statement of the
theorem more carefully, we would notice that -1 > -2, so that
there are no solutions to this equation.
- 1 = tan(x)
- Solution
- Since the table
says that
= sin(p/4)
= cos(p/4),
we can conclude that tan(p/4)
= sin(p/4)/cos(p/4)
= 1, so tan-1(1)
= p/4. According to formula for
tan-1,
the set of all solutions to 1 = tan(x)
is then x = kp
+ tan-1(1) or x = kp
+ p/4. As before, if we
simply use a calculator, we would obtain the approximate value of
0.7854 in place of p/4.
- -
= tan(x)
- Solution
- Since the table
says that
= sin(p/3) and 0.5
= cos(p/3),
we can conclude that tan(p/3)
= sin(p/3)/cos(p/3)
= .
Since tan is odd,
we know that -
= tan(-p/3), so tan-1(-)
= -p/3. According to formula for
tan-1,
the set of all solutions to -
= tan(x)
is then x = kp
+ tan-1(-)
or x = kp
- p/3. If we simply use a
calculator, we would obtain the approximate value of
1.0472 in place of p/3.
- 0.6 = cos(x)
- Solution
- The formula for
cos-1 gives
the set of all solutions to 0.6 = cos(x)
as x = 2kp
± cos-1(0.6) » 2kp
± 0.9273.
Back to Exercises.
Solving Trigonometric Equations
- Use the formulas from the text for sin-1,
cos-1, and tan-1 to write formulas for all solutions
to the following equations. Note: If you recognize exact values using the table
in the text, you may give exact values for these inverse trigonometric
functions. Otherwise, you should simply use your calculator to give approximate values.
- 4 = -5sin(x) + 1
- Solution
- Subtracting 1 from both sides, then dividing by -5, gives -0.6
= sin(x). The formula for sin-1
then gives the set of all solutions as x = 2kp
+ sin-1(-0.6), (2k + 1)p - sin-1(-0.6)
or x » 2kp
- 0.6435, (2k + 1)p + 0.6435.
- 4 = -5sin(1 - 2x) + 1
- Solution
- Proceeding as before (i.e., subtracting 1 from both sides, then
dividing by -5), gives 1 - 2x » 2kp
- 0.6435, (2k + 1)p + 0.6435.
Subtracting 1 from both sides, then dividing by -2 then gives x
» (2kp
- 0.6435 - 1)/(-2), ((2k + 1)p
+ 0.6435 - 1)/(-2) = -kp
+ 0.82175, -(k + 1/2)p
+ 0.17825.
- -7 = 2cos(x) - 3
- Solution
- Adding 3 to both sides, then dividing by 2, gives -2 =
cos(x), which, as we have already seen in the previous
Exercise, has no solution.
- 5 = 3tan(x) + 2
- Solution
- Subtracting 2 from both sides, then dividing by 3, gives 1
= tan(x), which, as we have already seen in the previous
Exercise, has solutions x = kp
+ p/4 or x
» kp
+ 0.7854.
- 5 = 3tan(-4x + p/3)
+ 2
- Solution
- Subtracting 2 from both sides, then dividing by 3, and using the formula
for
tan-1 gives -4x + p/3
= kp + p/4
or x » kp
+ 0.7854. Subtracting p/3
from both sides, then dividing by -4, gives x
= -kp/4 + p/48
or x » -kp/4
+ 0.06545.
- 1 = -5cos(3x - p/5)
+ 4
- Solution
- Subtracting 4 from both sides, then dividing by -5, and using the formula for
cos-1 gives 3x - p/5
» 2kp
± 0.9273. Adding p/5 to
both sides, then dividing by 3, gives x »
2kp/3 + 0.5185 or 2kp/3
- 0.09966.
Back to Exercises.
- Give the specific solution(s) requested to each of the
following equations.
- The first solution of 4 = -5sin(1 - 2x) + 1
that is less than -6.
- Solution
- Using our formula from the previous Exercise,
x
» -kp
+ 0.82175, -(k + 1/2)p
+ 0.17825, and plugging in k = -1,
0, 1, gives the values 3.96334, 1.74905, 0.82175, -1.39255,
-2.31984, and -4.53414. From this we can estimate that the
desired solution occurs when k is
around 2. Plugging in k = 2 and 3,
gives the values -5.46144, -7.67573, -8.60303, and -10.8173, so that
the desired solution is -7.67573.
- The first solution of 5 = 3tan(-4x + p/3)
+ 2 that is greater than 4.
- Solution
- Using our formula from the previous Exercise,
x » -kp/4
+ 0.06545, and plugging in k = -1,
0, 1, gives the values 0.850848, 0.06545, and -0.719948.
From this we can estimate that the desired solution occurs when k
is around -4 or -5. Plugging in k =
-6, -5 and -4, gives the values 4.77784, 3.99244, 3.20704, so
that the desired solution is 4.77784.
- The solutions of 1 = -5cos(3x - p/5)
+ 4
that are between p/2 and p.
- Solution
- Using our formula from the previous Exercise, x »
2kp/3 + 0.5185, 2kp/3
- 0.09966, and plugging in k = -1,
0, 1, gives the values -1.5759, -2.19406, 0.5185, -0.09966,
2.6129, and 1.99474. Since p/2 »
1.5 and p »
3, this we can estimate that the desired solutions occurs when k
is around 1. Plugging in k =
1 and 2, gives the values 2.6129, 1.99474, 4.70729, and
4.08913, so that the only solutions between p/2
and p are 2.6129 and 1.99474.
Back to Exercises.
Go to Geometry and Trigonometric
Functions.
Send
questions or comments to jwicks@northpark.edu