Download - 9.4 Evaluate Inverse Trigonometric Functions
9.4 Evaluate Inverse Trigonometric Functions
How are inverse Trigonometric functions used?How much information must be given about side lengths in a right triangle in order for you to be able to find the measures
of its acute angles?
Inverse Trig Functions•
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Inverse Trig Functions•
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Inverse Trig Functions•
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Evaluate the expression in both radians and degrees.
a. cos–1 32
√
SOLUTION
a. When 0 θ π or 0° 180°, the angle whose cosine is
≤ ≤ ≤ θ ≤32
√
cos–1 32
√θ =π6
= cos–1 32
√θ = = 30°
0°360°180°
90°
270°
45°135°
225° 315°
30°
60°120°
150°
210°
240° 300°
330°
x
y
Evaluate the expression in both radians and degrees.
b. sin–1 2
SOLUTION
sin–1b. There is no angle whose sine is 2. So, is undefined.
2
Evaluate the expression in both radians and degrees.
3 ( – )c. tan–1 √
SOLUTION
c. When – < θ < , or – 90° < θ < 90°, the angle whose tangent is – is:
π2
π2
√ 3
( – )tan–1 3√θ =π3
–= ( – )tan–1 3√θ = –60° =
Evaluate the expression in both radians and degrees.
1. sin–1 22
√
ANSWERπ4
, 45°
2. cos–1 12
ANSWER π3
, 60°
3. tan–1 (–1)
ANSWER π4
, –45°–
4. sin–1 (– )12
π6
, –30°–ANSWER
Solve the equation sin θ = – where 180° < θ < 270°.
58
SOLUTIONSTEP 1
sine is – is sin–1 – 38.7°. This58
58
–
Use a calculator to determine that in theinterval –90° θ 90°, the angle whose≤ ≤
angle is in Quadrant IV, as shown.
STEP 2 Find the angle in Quadrant III (where180° < θ < 270°) that has the same sinevalue as the angle in Step 1. The angle is:
θ 180° + 38.7° = 218.7°CHECK : Use a calculator to check the answer.
58sin
218.7°– 0.625=–
Solve a Trigonometric Equation
Solve the equation for
270° < θ < 360°5. cos θ = 0.4;
ANSWER about 293.6°
180° < θ < 270°6. tan θ = 2.1;
ANSWER about 244.5°
270° < θ < 360°7. sin θ = –0.23;
ANSWER about 346.7°
6.2934.66360
5.2441805.64
7.3463.13360
180° < θ < 270°8. tan θ = 4.7;
ANSWER about 258.0°
90° < θ < 180°9. sin θ = 0.62;
ANSWER about 141.7°
180° < θ < 270°10. cos θ = –0.39;
ANSWER about 247.0°
Solve the equation for
25818078
7.1413.38180
247113360
SOLUTION
In the right triangle, you are given the lengths of the side adjacent to θ and the hypotenuse, so use the inverse cosine function to solve for θ.
cos θ =adjhyp =
611
cos – 1θ = 611
56.9°
The correct answer is C.ANSWER
Monster Trucks
A monster truck drives off a ramp in order to jump onto a row of cars. The ramp has a height of 8 feet and a horizontal length of 20 feet. What is the angle θ of the ramp?
http://www.youtube.com/watch?v=SrzXaDFZcAo
http://www.youtube.com/watch?v=7SjX7A_FR6g
SOLUTION
STEP 1 Draw: a triangle that represents the ramp.
STEP 2 Write: a trigonometric equation that involves the ratio of the ramp’s height and horizontal length.
tan θ =oppadj =
820
STEP 3 Use: a calculator to find the measure of θ.
tan–1θ = 820
21.8°
The angle of the ramp is about 22°.
ANSWER
Find the measure of the angle θ.
11.
SOLUTION
In the right triangle, you are given the lengths of the side adjacent to θ and the hypotenuse. So, use the inverse cosine function to solve for θ.
cos θ =adjhyp = 4
9= 63.6°θ cos–1 4
9
Find the measure of the angle θ.
SOLUTION
In the right triangle, you are given the lengths of the side opposite to θ and the side adjacent. So, use the inverse tan function to solve for θ.
12.
tan θ =oppadj =
108
θ 51.3°= tan–1 108
Find the measure of the angle θ.
SOLUTION
In the right triangle, you are given the lengths of the side opposite to θ and the hypotenuse. So, use the inverse sin function to solve for θ.
13.
sin θ =opphyp = 5
1224.6°θ = sin–1 5
12
9.4 AssignmentPage 582, 3-29 odd