Transcript monster truck

Lesson 13.4, For use with pages 875-880
Evaluate the expression.
1. cos 45º
2
ANSWER
2
2. sin
5π
6
ANSWER
1
2
3. tan(– 60º)
ANSWER
– 3
Lesson 13.4, For use with pages 875-880
Evaluate the expression.
4. cos π
–1
ANSWER
5.
π
tan – 6
ANSWER
–
3
3
Trigonometry, Inverse Functions
EXAMPLE 1
Evaluate inverse trigonometric functions
Evaluate the expression in both radians and degrees.
a.
√ 3
cos–1
2
SOLUTION
a.
When 0 ≤θ ≤ π or 0°
is √ 3
≤ θ ≤ 180°,
the angle whose cosine
2
θ = cos–1
√ 3
2
=
π
6
θ = cos–1
√ 3
2
= 30°
EXAMPLE 1
Evaluate inverse trigonometric functions
Evaluate the expression in both radians and degrees.
b.
sin–1
2
SOLUTION
b.
There is no angle whose sine is 2.
undefined.
So,
sin–1 2 is
EXAMPLE 1
Evaluate inverse trigonometric functions
Evaluate the expression in both radians and degrees.
c.
tan–1 ( – √
3
SOLUTION
c.
π
π
<θ<
, or – 90° < θ < 90°, the
2
2
angle whose tangent is is:
√ 3
When –
θ =
tan–1
( – √ 3)
= –
π
3
θ = tan–1 ( – √ 3)
= –60°
EXAMPLE 2
Solve a trigonometric equation
Solve the equation sin θ = –
5
where 180°
8 < θ < 270°.
SOLUTION
STEP 1
Use a calculator to determine that in the
interval –90° ≤ θ ≤ 90°, the angle whose
5
5
sine is –
is sin–1 –
– 38.7°. This
8
8
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
CHECK :
180° + 38.7° = 218.7°
Use a calculator to check the answer.
5

sin 218.7°
– 0.625 = –
8
for Examples 1 and 2
GUIDED PRACTICE
Evaluate the expression in both radians and degrees.
1.
sin–1
√ 2
2
π , 45°
4
ANSWER
2.
cos–1
1
2
π , 60°
3
ANSWER
3.
tan–1 (–1)
ANSWER
–
π , –45°
4
for Examples 1 and 2
GUIDED PRACTICE
Evaluate the expression in both radians and degrees.
4.
sin–1 (– )
ANSWER
1
2
–
π , –30°
6
GUIDED PRACTICE
for Examples 1 and 2
Solve the equation for
5. cos θ = 0.4;
ANSWER
6. tan θ = 2.1;
ANSWER
7. sin θ = –0.23;
ANSWER
270° < θ < 360°
about 293.6°
180° < θ < 270°
about 244.5°
270° < θ < 360°
about 346.7°
GUIDED PRACTICE
for Examples 1 and 2
Solve the equation for
8. tan θ = 4.7;
ANSWER
9. sin θ = 0.62;
ANSWER
10. cos θ = –0.39;
ANSWER
180° < θ < 270°
about 258.0°
90° < θ < 180°
about 141.7°
180° < θ < 270°
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
=
ANSWER
adj
hyp
6
=
11
The correct answer is C.
θ = cos
–1
6
11
56.9°
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
STEP 3
Write and solve a trigonometric equation
Use: a calculator to find the measure of θ.
θ =
tan–1
8
20
ANSWER
The angle of the ramp is about 22°.
21.8°
for Examples 3 and 4
GUIDED PRACTICE
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
=
4
9
θ = cos–1
4
9
63.6°
for Examples 3 and 4
GUIDED PRACTICE
Find the measure of the angle θ.
12.
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.
tan θ
=
opp
adj
=
10
8
θ = tan–1
10
8
51.3°
for Examples 3 and 4
GUIDED PRACTICE
Find the measure of the angle θ.
13.
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.
sin θ
=
opp
hyp
=
5
12
θ = sin–1
5
12
24.6°
for Examples 3 and 4
GUIDED PRACTICE
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
for Examples 3 and 4
GUIDED PRACTICE
STEP 3
Use: a calculator to find the measure of θ.
θ =
tan–1
10
26
ANSWER
The angle of the ramp is about 22.6°.
22.6°