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?
Feb 23, 2016
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•
x
y
Inverse Trig Functions•
x
y
0
Inverse Trig Functions•
x
y
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