1. Radians and Degrees Trigonometry MATH 103 S. Rook

2. Overview Section 3.2 in the textbook: – The radian – Converting between degrees and radians – Radians and Trigonometric functions – Approximating with a calculator 2

3. The Radian

4. Motivation for Introducing Radians Thus far we have worked exclusively with angles measured in degrees In some calculations, we require theta 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 4

5. Radians 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 – Informally: How many radii r comprise the arc length s For θ = 1 radian, s = r 5

6. Radians (Example) Ex 1: Find the radian measure of θ, if θ is a central angle in a circle of radius r, and θ cuts off an arc of length s: r = 10 inches, s = 5 inches 6

7. Converting Between Degrees and Radians

8. 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 8

9. 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 9

10. Converting from Degrees to Radians 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 10

11. Converting from Degrees to Radians (Example) Ex 2: For each angle θ, i) draw θ in standard position, ii) convert θ to radian measure using exact values, iii) name the reference angle in both degrees and radians: a) -120° b)390° 11

12. Converting from Radians to Degrees To convert from radians to degrees: – Multiply by the conversion ratio so that radians will divide out leaving degrees The concept of reference angles still applies when θ is in radians: – Instead of adding/subtracting 180° or 360°, add/subtract π or 2π respectively 12

13. Converting from Radians to Degrees (Example) Ex 3: For each angle θ, i) draw θ in standard position, ii) convert θ to degree measure, iii) name the reference angle in both radians and degrees: a) b) 13

14. Radians and Trigonometric Functions

15. On the next slide is a table of values you should have already memorized when θ was in degrees Only difference is the equivalent radian values – Each radian value can be obtained via the conversion procedure previously discussed 15

16. Radians and Trigonometric Functions (Continued) DegreesRadianscos θsin θ 0°010 30° 45° 60° 90°01 180°0 270°0 360°10 16

17. Radians and Trigonometric Functions (Example) Ex 4: Give the exact value: a) b) 17

18. Radians and Trigonometric Functions (Example) Ex 5: Find the value of y that corresponds to each value of x and then write each result as an ordered pair (x, y): y = cos x for 18

19. Approximating with a Calculator

20. When approximating the value of a trigonometric function in radians: – Ensure that the calculator is set to radian mode ESSENTIAL to know when to use degree mode and when to use radian mode: – Angle measurements in degrees are post-fixed with the degree symbol (°) – Angle measurements in radians are sometimes given the post-fix unit rad but more commonly are given with no units at all 20

21. Approximating with a Calculator (Example) Ex 6: Use a calculator to approximate: a)sin 1 b)cos 3π c) 21

22. Summary After studying these slides, you should be able to: – Calculate the radian measure of a circle with radius r cutting off an arc length of s – Convert between degrees & radians and vice versa – Apply previous concepts such as reference angles to angles measured in radians – Use a calculator to approximate the value of a trigonometric function in both degrees and radians Additional Practice – See the list of suggested problems for 3.2 Next lesson – Definition III: Circular Functions (Section 3.3) 22