6. You have evaluated trigonometric functions for a
given angle. You can also find the measure of
angles given the value of a trigonometric function
by using an inverse trigonometric relation.
7. The expression sin-1
is read as “the inverse sine.”
In this notation,-1
indicates the inverse of the sine
function, NOT the reciprocal of the sine function.
Reading Math
8. The inverses of the trigonometric
functions are not functions
themselves because there are
many values of θ for a particular
value of a. For example, suppose
that you want to find cos-1
.
Based on the unit circle, angles
that measure and radians
have a cosine of . So do all
angles that are coterminal with
these angles.
9. Example 1: Finding Trigonometric Inverses
Find all possible values of cos-1
.
Step 1 Find the values between
0 and 2π radians for which cos θ
is equal to .
Use the x-coordinates of points on the
unit circle.
10. Example 1 Continued
Step 2 Find the angles that are coterminal
with angles measuring and radians.
Add integer multiples of
2π radians, where n is
an integer
Find all possible values of cos-1
.
11. Check It Out! Example 1
Find all possible values of tan-1
1.
Step 1 Find the values between
0 and 2π radians for which tan θ
is equal to 1.
Use the x and y-coordinates of
points on the unit circle.
12. Add integer multiples of
2π radians, where n is
an integer
Check It Out! Example 1 Continued
Find all possible values of tan-1
1.
Step 2 Find the angles that are coterminal
with angles measuring and radians.
13. Because more than one value of θ produces the
same output value for a given trigonometric
function, it is necessary to restrict the domain of
each trigonometric function in order to define the
inverse trigonometric functions.
14. Trigonometric functions with restricted domains
are indicated with a capital letter. The domains of
the Sine, Cosine, and Tangent functions are
restricted as follows.
Sinθ = sinθ for {θ| }
θ is restricted to Quadrants I and IV.
Cosθ = cosθ for {θ| }
θ is restricted to Quadrants I and II.
Tanθ = tanθ for {θ| }
θ is restricted to Quadrants I and IV.
15. These functions can be used to define the inverse
trigonometric functions. For each value of a in the
domain of the inverse trigonometric functions,
there is only one value of θ. Therefore, even
though tan-1
has many values, Tan-1
1 has only
one value.
17. The inverse trigonometric functions are also
called the arcsine, arccosine, and arctangent
functions.
Reading Math
18. Example 2A: Evaluating Inverse Trigonometric
Functions
Evaluate each inverse trigonometric function.
Give your answer in both radians and degrees.
Find value of θ for
or whoseCosine .
Use x-coordinates of points on the
unit circle.
19. Example 2B: Evaluating Inverse Trigonometric
Functions
Evaluate each inverse trigonometric function.
Give your answer in both radians and degrees.
The domain of the inverse sine function is
{a|1 = –1 ≤ a ≤ 1}. Because is outside this
domain. Sin-1
is undefined.
20. Check It Out! Example 2a
Evaluate each inverse trigonometric function.
Give your answer in both radians and degrees.
Find value of θ for
or whoseSine is .
Use y-coordinates of points
on the unit circle.
21. Check It Out! Example 2b
Evaluate each inverse trigonometric function.
Give your answer in both radians and degrees.
Find value of θ for
or whoseCosine is 0.
Use x-coordinates of points on the
unit circle.
0 = Cos θ
(0, 1)
22. Example 3: Safety Application
A painter needs to lean a 30 ft ladder against
a wall. Safety guidelines recommend that the
distance between the base of the ladder and
the wall should be of the length of the
ladder. To the nearest degree, what acute
angle should the ladder make with the
ground?
23. Example 3 Continued
θ
7.5
Step 1 Draw a diagram. The base of the ladder
should be (30) = 7.5 ft from the wall. The angle
between the ladder and the ground θ is the
measure of an acute angle of a right triangle.
24. Example 3 Continued
Step 2 Find the value of θ.
Use the cosine ratio.
Substitute 7.5 for adj. and
30 for hyp. Then
simplify.
The angle between the ladder
and the ground should be about
76°
25. Check It Out! Example 3
A group of hikers wants to walk form a lake
to an unusual rock formation.
The formation is 1 mile east and 0.75 mile
north of the lake. To the nearest degree, in
what direction should the hikers head from
the lake to reach the rock formation?
Step 1 Draw a diagram. The
base of the triangle should be 1
mile. The angle North from that
point to the rock is 0.75 miles.
θ is the measure of an acute
angle of a right triangle.
Lake
θ
Rock
0.75 mi
1 mi
26. Check It Out! Example 3 Continued
Step 2 Find the value of θ
Use the tangent ratio.
Substitute 0.75 for opp.
and 1 for adj. Then
simplify.
The angle the hikers should
take is about 37° north of east.
27. Example 4A: Solving Trigonometric Equations
Solve each equation to the nearest tenth. Use
the given restrictions.
sin θ = 0.4, for – 90° ≤ θ ≤ 90°
The restrictions on θ are the same as those for
the inverse sine function.
θ = Sin-1
(0.4) ≈ 23.6°
Use the inverse sine
function on your
calculator.
28. Example 4B: Solving Trigonometric Equations
Solve each equation to the nearest tenth. Use
the given restrictions.
sin θ = 0.4, for 90° ≤ θ ≤ 270°
The terminal side of θ is
restricted to Quadrants ll
and lll. Since sin θ > 0,
find the angle in Quadrant
ll that has the same sine
value as 23.6°.
θ ≈ 180° –23.6° ≈ 156.4°
θ has a reference
angle of 23.6°, and
90° < θ < 180°.
29. Check It Out! Example 4a
Solve each equations to the nearest tenth.
Use the given restrictions.
tan θ = –2, for –90° < θ < 90°
The restrictions on θ are the same for those of
the inverse tangent function.
θ = Tan-1
–2 ≈ –63.4° Use the inverse tangent
function on your
calculator.
30. Check It Out! Example 4b
Solve each equations to the nearest tenth.
Use the given restrictions.
tan θ = –2, for 90° < θ < 180°
The terminal side of θ is restricted to Quadrant II.
Since tan θ < 0, find the angle in Quadrant II
that has the same value as –63.4°.
θ ≈ 180° – 63.4° ≈ 116.6°
116.6°
–63.4°
31. Lesson Quiz: Part I
1. Find all possible values of cos-1
(–1).
2. Evaluate Sin-1
Give your answer in both radians
and degrees.
3. A road has a 5% grade, which means that there is a
5 ft rise for 100 ft of horizontal distance. At what
angle does the road rise from the horizontal? Round
to the nearest tenth of a degree.
2.9°
32. Lesson Quiz: Part II
Solve each equation to the nearest tenth.
Use the given restrictions.
4. cos θ = 0.3, for 0° ≤ θ ≤ 180°
5. cos θ = 0.3, for 270° < θ < 360° θ ≈ 287.5°
θ ≈ 72.5°
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