Transcript Holt McDougal Algebra 2 10-4

Inverses
InversesofofTrigonometric
Trigonometric
10-4Functions
10-4
Functions
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
Algebra 2Algebra 2
Holt
Inverses of Trigonometric
10-4 Functions
Warm Up
Convert each measure from degrees to
radians.
1. 120°
2. 180°
3. 225°
4. –30°
Holt McDougal Algebra 2
Inverses of Trigonometric
10-4 Functions
Warm Up
Find the exact value of each trigonometric
function.
5.
6.
7.
8.
Holt McDougal Algebra 2
Inverses of Trigonometric
10-4 Functions
Objectives
Evaluate inverse trigonometric
functions.
Use trigonometric equations and
inverse trigonometric functions to
solve problems.
Holt McDougal Algebra 2
Inverses of Trigonometric
10-4 Functions
Vocabulary
inverse sine functions
inverse cosine function
inverse tangent function
Holt McDougal Algebra 2
Inverses of Trigonometric
10-4 Functions
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.
Holt McDougal Algebra 2
Inverses of Trigonometric
10-4 Functions
Reading Math
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.
Holt McDougal Algebra 2
Inverses of Trigonometric
10-4 Functions
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
have a cosine of
radians
. So do all
angles that are coterminal with
these angles.
Holt McDougal Algebra 2
Inverses of Trigonometric
10-4 Functions
Example 1: Finding Trigonometric Inverses
Find all possible values of cos-1
Step 1 Find the values between
0 and 2 radians for which cos θ
is equal to
.
Use the x-coordinates of points on the
unit circle.
Holt McDougal Algebra 2
.
Inverses of Trigonometric
10-4 Functions
Example 1 Continued
Find all possible values of cos-1
.
Step 2 Find the angles that are coterminal
with angles measuring
and
radians.
Add integer multiples of
2 radians, where n is
an integer
Holt McDougal Algebra 2
Inverses of Trigonometric
10-4 Functions
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.
Holt McDougal Algebra 2
Inverses of Trigonometric
10-4 Functions
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.
Holt McDougal Algebra 2
Inverses of Trigonometric
10-4 Functions
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-11 has
only one value.
Holt McDougal Algebra 2
Inverses of Trigonometric
10-4 Functions
Reading Math
The inverse trigonometric functions are also
called the arcsine, arccosine, and arctangent
functions.
Holt McDougal Algebra 2
Inverses of Trigonometric
10-4 Functions
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 whose
Cosine
.
Use x-coordinates of points on the
unit circle.
Holt McDougal Algebra 2
Inverses of Trigonometric
10-4 Functions
Check It Out! Example 2b
Evaluate each inverse trigonometric function.
Give your answer in both radians and degrees.
(0, 1)
0 = Cos θ

Find value of θ for
or whose
Cosine is 0.
Use x-coordinates of points on the
unit circle.
Holt McDougal Algebra 2
Inverses of Trigonometric
10-4 Functions
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?
Holt McDougal Algebra 2
Inverses of Trigonometric
10-4 Functions
Example 3 Continued
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.
7.5
Holt McDougal Algebra 2
θ
Inverses of Trigonometric
10-4 Functions
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°
Holt McDougal Algebra 2
Inverses of Trigonometric
10-4 Functions
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.
Lake
θ is the measure of an acute
angle of a right triangle.
Holt McDougal Algebra 2
Rock
0.75 mi
θ
1 mi
Inverses of Trigonometric
10-4 Functions
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.
Holt McDougal Algebra 2
Inverses of Trigonometric
10-4 Functions
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.
 =
Sin-1(0.4)
Holt McDougal Algebra 2
≈ 23.6°
Use the inverse sine
function on your
calculator.
Inverses of Trigonometric
10-4 Functions
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 has a reference
angle of 23.6°, and
θ ≈ 180° –23.6° ≈ 156.4°
90° < θ < 180°.
Holt McDougal Algebra 2
Inverses of Trigonometric
10-4 Functions
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°
Holt McDougal Algebra 2
Use the inverse tangent
function on your
calculator.
Inverses of Trigonometric
10-4 Functions
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°.
116.6°
–63.4°
θ ≈ 180° – 63.4° ≈ 116.6°
Holt McDougal Algebra 2
Inverses of Trigonometric
10-4 Functions
Lesson Quiz: Part I
1. Find all possible values of cos-1(–1).
2. Evaluate Sin-1
and degrees.
Give your answer in both radians
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°
Holt McDougal Algebra 2
Inverses of Trigonometric
10-4 Functions
Lesson Quiz: Part II
Solve each equation to the nearest tenth.
Use the given restrictions.
4. cos θ = 0.3, for 0° ≤ θ ≤ 180° θ ≈ 72.5°
5. cos θ = 0.3, for 270° < θ < 360° θ ≈ 287.5°
Holt McDougal Algebra 2