1. Radian & Degree Measure
MATH Precalculus
S. Rook

2. Overview Section 4.1 in the textbook: Angles Degree measure
Radian measure
Converting between degrees & radians

3. Angles

4. Angles
Angle: describes the “space” between two rays that are joined at a common endpoint
Recall from Geometry that a ray has one terminating side and one non-terminating side
Can also think about an angle as a rotation about the common endpoint
Start at OA (Initial side)
End at OB (Terminal side)

5. Angles (Continued) If the initial side is rotated counter-clockwise
θ is a positive angle
clockwise
θ is a negative angle

6. Angles in Standard Position
An angle θ is in standard position if its:
Initial side extends along the positive x-axis
in reference to the Cartesian Plane
Vertex is (0, 0)
The “element of” symbol can be used to denote an angle in standard position
e.g means θ is in standard position with its terminal side in Quadrant III

7. Degree Measure

8. Angle Measure Angle Measure: expresses the size of an angle
i.e. the space in between the initial and terminal sides in the direction of rotation
Two common types of angle measures:
Degrees
Radians

9. Degree Measure
1 degree corresponds to (1⁄360) of a complete revolution starting from the initial side of an angle to its terminal side
i.e. Can be viewed in terms of a circle
Common degree measurements to be familiar with:
360° makes one complete revolution
The initial and terminal sides of the angle are the same
180° makes one half of a complete revolution
90° makes one quarter of a complete revolution

10. Degree Measure (Continued)
Angles that measure:
Between 0° and 90° are known as acute angles
Exactly 90° are known as right angles
Denoted by a small square between the initial and terminal sides
Between 90° and 180° are known as obtuse angles
Complementary angles: two angles whose measures sum to 90°
Supplementary angles: two angles whose measures sum to 180°

11. Degrees & Minutes
Degrees can be broken down even further using minutes
1° = 60’
To convert from decimal degrees to degrees and minutes:
Use the decimal portion of the angle
Multiply by the appropriate conversion ratio
Align the units in the ratio so the degrees will divide out, leaving the minutes
To convert from degrees and minutes to decimal degrees:
Use the minutes from the angle measurement
Align the units in the ratio so the minutes will divide out, leaving the degrees

12. Sketching Angles in Standard Position (Example)
Ex 1: Sketch each angle in standard position: a) 293° b) -115°

13. Complementary & Supplementary Angles (Example)
Ex 2: Find: i) the complement ii) the supplement θ = 65°

14. Converting from Degrees to Minutes & Vice Versa (Example)
Ex 3: Convert a) to degrees and minutes and convert b) to decimal degrees – approximate if necessary: a) θ = ° b) θ = 17° 22’

15. Radian Measure

16. Motivation for Introducing Radians
In some calculations, we require the measure of an angle (θ) to be a real number – we need a unit other than degrees
This unit is known as the radian
Many calculations tend to become easier to perform when θ is in radians
Further, some calculations can be performed or even simplified ONLY if θ is in radians
However, degrees are still in use in many applications so a knowledge of both degrees and radians is ESSENTIAL

17. Radians For θ = 1 radian, s = r
Radian Measure: A circle with central angle θ and radius r which cuts off an arc of length s has a central angle measure of where θ is in radians
i.e. How many radii r comprise the arc length s
For θ = 1 radian,
s = r

18. Radian Measure (Example)
Ex 4: Find the radian measure of the central angle of a circle of radius r that subtends an arc length of s A radius of 27 inches and an arc length of 6 inches

19. Converting Between Degrees and Radians

20. Relationship Between Degrees and Radians
Given a circle with radius r, what arc length s is required to make one complete revolution?
Recall that the circumference measures the distance or length around a circle
What is the circumference of a circle with radius r?
C = 2πr
Thus, s = 2πr is the arc length of one revolution and
is the number of radians in one revolution
Therefore, θ = 360° = 2π consists of a complete revolution around a circle

21. Relationship Between Degrees and Radians (Continued)
Equivalently: 180° = π radians
You MUST memorize this conversion!!!
Technically, when measured in radians, θ is unitless, but we sometimes append “radians” to it to differentiate radians from degrees
Like radians, real numbers are unitless as well

22. Converting from Degrees to Radians & Vice Versa
To convert from degrees to radians:
Multiply by the conversion ratio so that degrees will divide out leaving radians
If an exact answer is desired, leave π in the final answer
If an approximate answer is desired, use a calculator to estimate π
To convert from radians to degrees:
Multiply by the conversion ratio so that radians will divide out leaving degrees

23. Common Angles
Need to become familiar with the degree and radian conversion between the following commonly used angle measurements:
Deg
Rad 0°
30°
π⁄6
45°
π⁄4
60°
π⁄3
90°
π⁄2
180° π
270°
3π⁄2
360° 2π

24. Converting from Degrees to Radians & Vice Versa (Example)
Ex 5: Convert a) & b) to degrees and convert c) & d) to radians – leave π in the answer when necessary: a) b) c) d)

25. Coterminal Angles Two angles are coterminal if:
BOTH are standard angles
Share the SAME terminal side
How can we obtain an angle coterminal to an angle θ?
The second angle must terminate where θ terminates
Recall that one complete revolution around a circle is 360° in degrees or 2π in radians

26. Coterminal Angles (Example)
Ex 6: Do the following: a) Given θ = -190° find in degrees: i) two coterminal angles and ii) all angles coterminal to θ b) Given θ = π⁄8 find in radians: i) two coterminal angles and ii) all angles coterminal to θ

27. Summary After studying these slides, you should be able to:
Draw an angle in standard position
Find both the complement and supplement of an angle
Convert between degrees & minutes and decimal degrees and vice versa
Calculate the radian measure of a circle with radius r and subtended by an arc length s
Convert between radians & degrees and vice versa
Additional Practice
See the list of suggested problems for 4.1
Next lesson
Trigonometric Functions: The Unit Circle (Section 4.2)