Five-Minute Check (over Lesson 4-1)
Then/Now
New Vocabulary
Example 1: Convert Between DMS and Decimal Degree Form
Key Concept: Radian Measure
Key Concept: Degree/Radian Conversion Rules
Example 2: Convert Between Degree and Radian Measure
Key Concept: Coterminal Angles
Example 3: Find and Draw Coterminal Angles
Key Concept: Arc Length
Example 4: Find Arc Length
Key Concept: Linear and Angular Speed
Example 5: Real-World Example: Find Angular and Linear Speeds
Key Concept: Area of a Sector
Example 6: Find Areas of Sectors
Over Lesson 4-1
Find the exact values of the six
trigonometric functions of θ.
A.
B.
C.
D.
Over Lesson 4-1
If
, find the exact values of the five
remaining trigonometric functions of θ.
A.
B.
C.
D.
Over Lesson 4-1
Solve ΔABC. Round side lengths to the nearest
tenth and angle measures to the nearest degree.
A. a ≈ 26.8, c ≈ 13.8, C = 31o
B. a ≈ 19.7, c ≈ 11.8, C = 31o
C. a ≈ 11.8, c ≈ 19.7, C = 31o
D. a ≈ 15.1, c ≈ 17.3, C = 41o
Over Lesson 4-1
Find the value of x. Round to the nearest tenth.
A. 37.1
B. 32.5
C. 15.7
D. 8.7
Over Lesson 4-1
If
A.
B.
C.
D.
, find tan θ.
You used the measures of acute angles in triangles
given in degrees. (Lesson 4-1)
• Convert degree measures of angles to radian
measures, and vice versa.
• Use angle measures to solve real-world problems.
• vertex
• initial side
• terminal side
• standard position
• radian
• coterminal angles
• linear speed
• angular speed
• sector
Convert Between DMS and Decimal Degree
Form
A. Write 329.125° in DMS form.
First, convert 0.125° into minutes and seconds.
329.125° = 329° +
= 329° + 7.5'
1° = 60'
Simplify.
Next, convert 0.5' into seconds.
329.125° = 329° + 7' +
= 329° + 7' + 30"
1' = 60"
Simplify.
Therefore, 329.125° can be written as 329°7'30".
Answer: 329°7'30"
Convert Between DMS and Decimal Degree
Form
B. Write 35°12'7'' in decimal degree form to the
nearest thousandth.
Each minute is
of a degree and each second is
of a minute, so each second is
35°12'7" = 35o + 12'
of a degree.
Convert Between DMS and Decimal Degree
Form
≈ 35° + 0.2 + 0.002
Simplify.
≈ 35.202°
Add.
Therefore, 35°12'7" can be written as about 35.202°.
Answer: 35.202°
Write 141.275° in DMS form.
A. 141°12'4.5"
B. 141.2°45'0"
C. 141°4'35"
D. 141°16'30"
Convert Between Degree and Radian Measure
A. Write 135° in radians as a multiple of π.
Answer:
Convert Between Degree and Radian Measure
B. Write –30° in radians as a multiple of π.
Answer:
Convert Between Degree and Radian Measure
C. Write
in degrees.
= 120°
Answer: 120°
Simplify.
Convert Between Degree and Radian Measure
D. Write
in degrees.
= 135°
Answer: –135°
Simplify.
Write 150o in radians as a multiple of π.
A.
B.
C.
D.
Find and Draw Coterminal Angles
A. Identify all angles that are coterminal with 80°.
Then find and draw one positive and one negative
angle coterminal with 80°.
All angles measuring 80° + 360n° are coterminal with
an 80° angle. Let n = 1 and –1.
80° + 360(1)°= 80° + 360° or 440°
Find and Draw Coterminal Angles
80° + 360(–1)° = 80° – 360° or –280°
Answer: 80o + 360no; Sample answers: 440o, –280o
Find and Draw Coterminal Angles
B. Identify all angles that are coterminal with
.
Then find and draw one positive and one negative
angle coterminal with
.
All angles measuring
angle. Let n = 1 and –1.
are coterminal with a
Find and Draw Coterminal Angles
Answer:
Sample answer:
Identify one positive and one negative angle
coterminal with a 126o angle.
A. 486°, –234°
B. 54°, –126°
C. 234°, –54°
D. 36°, –486°
Find Arc Length
A. Find the length of the intercepted arc in a
circle with a central angle measure of
and a
radius of 4 inches. Round to the nearest tenth.
Arc Length
r = 4 and
Simplify.
Find Arc Length
The length of the intercepted arc is
inches.
Answer: 4.2 in.
or about 4.2
Find Arc Length
B. Find the length of the intercepted arc in a circle
with a central angle measure of 125° and a radius
of 7 centimeters. Round to the nearest tenth.
Method 1
Convert 125o to radian measure, and then
use s = rθ to find the arc length.
Find Arc Length
Substitute r = 7 and
s = r
.
Arc length
r = 7 and
Simplify.
Find Arc Length
Method 2
Use
to find the arc length.
Arc length
r = 7 and θ = 125°
Simplify.
The length of the intercepted arc is
15.3 centimeters.
Answer: 15.3 cm
or about
Find the length of the intercepted arc in a circle
with radius 6 centimeters and a central angle with
measure
.
A. 2.4 centimeters
B. 4.7 centimeters
C. 28.3 centimeters
D. 45°
Find Angular and Linear Speeds
A. RECORDS A typical vinyl record has a
diameter of 30 cm. When played on a turn table,
the record spins at
revolutions per minute.
Find the angular speed, in radians per minute, of a
record as it plays. Round to the nearest tenth.
Because each rotation measures 2π radians,
revolutions correspond to an angle of rotation
Find Angular and Linear Speeds
Angular speed
Therefore, the angular speed of the record is
or about 209.4 radians per minute.
Answer: 209.4 radians per minute
Find Angular and Linear Speeds
B. RECORDS A typical vinyl record has a
diameter of 30 cm. When played on a turn table,
the record spins at
revolutions per minute.
Find the linear speed at the outer edge of the
record as it spins, in centimeters per second.
A rotation of
of rotation
revolutions corresponds to an angle
Find Angular and Linear Speeds
Linear Speed
s = r
minute
Simplify.
Find Angular and Linear Speeds
Use dimensional analysis to convert this speed from
centimeters per minute to centimeters per second.
Therefore, the linear speed of the record is about 52.4
centimeters per second.
Answer: about 52.4 cm/s
CAROUSEL Find the angular speed of a carousel
in radians per minute if the diameter is 6 feet and it
rotates at a rate of 10 revolutions per minute.
A. 31.4 radians per minute
B. 62.8 radians per minute
C. 188.5 radians per minute
D. 377.0 radians per minute
Find Areas of Sectors
A. Find the area of the sector
of the circle.
The measure of the sector’s central angle is
the radius is 5 meters.
Area of sector
r = 5 and
, and
Find Areas of Sectors
Therefore, the area of the sector is
square meters.
Answer:
or about 29.5
Find Areas of Sectors
B. Find the area of the sector
of the circle.
Convert the central angle measure to radians.
Then use the radius of the sector to find the area.
Find Areas of Sectors
Area of sector
r = 8 and
Therefore, the area of the sector is
square feet.
Answer:
or about 33.5
Find the area of the sector of the circle.
A. 7.9 in2
B. 15.7 in2
C. 58.9 in2
D. 117.8 in2