View
225
Download
6
Category
Preview:
Citation preview
Lesson 13.4, For use with pages 875-880
1. cos 45º
ANSWER1
2
Evaluate the expression.
2. sin 5π
6
3. tan(– 60º)
ANSWER – 3
ANSWER2
2
ANSWER –1
Lesson 13.4, For use with pages 875-880
ANSWER3
3–
5. tan – π
6
Evaluate the expression.
4. cos π
Trigonometry, Inverse Functions
EXAMPLE 1 Evaluate inverse trigonometric functions
Evaluate the expression in both radians and degrees.
a. cos–1 3
2
√
SOLUTION
a. When 0 θ π or 0° 180°, the angle whose cosine is
≤ ≤ ≤ θ ≤ 3
2
√
cos–1 3
2
√θ =
π
6= cos–1 3
2
√θ = = 30°
EXAMPLE 1 Evaluate inverse trigonometric functions
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
EXAMPLE 1 Evaluate inverse trigonometric functions
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:
π
2
π2
√ 3
( – )tan–1 3√θ =π
3–= ( – )tan–1 3 √θ = –60° =
EXAMPLE 2 Solve a trigonometric equation
Solve the equation sin θ = – where 180° < θ < 270°.5
8
SOLUTION
STEP 1
sine is – is sin–1 – 38.7°. This5
8
5
8–
Use a calculator to determine that in the
interval –90° θ 90°, the angle whose≤ ≤
angle is in Quadrant IV, as shown.
EXAMPLE 2 Solve a trigonometric equation
STEP 2
Find the angle in Quadrant III (where
180° < θ < 270°) that has the same sine
value as the angle in Step 1. The angle is:
θ 180° + 38.7° = 218.7°
CHECK : Use a calculator to check the answer.
5
8sin 218.7° – 0.625 = –
GUIDED PRACTICE for Examples 1 and 2
Evaluate the expression in both radians and degrees.
1. sin–1 2
2
√
ANSWERπ4
, 45°
2. cos–1 12
ANSWERπ3
, 60°
3. tan–1 (–1)
ANSWERπ4
, –45°–
GUIDED PRACTICE for Examples 1 and 2
Evaluate the expression in both radians and degrees.
4. sin–1 (– )12
π6
, –30°–ANSWER
GUIDED PRACTICE for Examples 1 and 2
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°
GUIDED PRACTICE for Examples 1 and 2
Solve the equation for
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°
EXAMPLE 3 Standardized Test Practice
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 =adj
hyp=
6
11cos – 1θ =
6
1156.9°
The correct answer is C.ANSWER
EXAMPLE 4 Write and solve a trigonometric equation
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?
EXAMPLE 4 Write and solve a trigonometric equation
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 =opp
adj=
8
20
EXAMPLE 4 Write and solve a trigonometric equation
STEP 3 Use: a calculator to find the measure of θ.
tan–1θ = 8
2021.8°
The angle of the ramp is about 22°.
ANSWER
GUIDED PRACTICE for Examples 3 and 4
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 =adj
hyp=
49
= 63.6°θ cos–1 49
GUIDED PRACTICE for Examples 3 and 4
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 =opp
adj=
108
θ 51.3°= tan–1 108
GUIDED PRACTICE for Examples 3 and 4
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 =opp
hyp=
512
24.6°θ = sin–1 512
GUIDED PRACTICE for Examples 3 and 4
14. WHAT IF? In Example 4, suppose a monster truck drives 26 feet on a ramp before jumping onto a row of cars. If the ramp is 10 feet high, what is the angle θ of the ramp?
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 =opp
adj=
10
26
GUIDED PRACTICE for Examples 3 and 4
STEP 3 Use: a calculator to find the measure of θ.
22.6°tan–1θ =10
26
The angle of the ramp is about 22.6°.
ANSWER
Recommended