Rev.S08

1. Review 5.1-5.3 Radian Measure and Circular Functions Rev.S08

2. How to Convert Between Degrees and Radians? • 1. Multiply a degree measure by radian and simplify to convert to radians. • 2. Multiply a radian measure by and simplify to convert to degrees. 2

3. Example of Converting from Degrees to Radians • Convert each degree measure to radians. • a) 60 • b) 221.7 OR 3

4. Example of Converting from Radians to Degrees • Convert each radian measure to degrees. • a)‏ • b) 3.25 4

5. Let’s Look at Some Equivalent Angles in Degrees and Radians Degrees Radians Degrees Radians Exact Approximate Exact Approximate 0 0 0 90 1.57 30 .52 180 π 3.14 45 .79 270 4.71 60 1.05 360 2π 6.28 Rev.S08 5

6. Let’s Look at Some Equivalent Angles in Degrees and Radians (cont.)‏ 6

7. Examples • Find each function value. • a)‏ • b) IF YOUR ANGLE IS ON THE UNIT CIRLE THEN USE IT!! Or you can set up your triangles! tan = opp adj sin = opp hyp θ θ 1 1 7

8. How to Find Arc Length of a Circle? • The length s of the arc intercepted on a circle of radius rby a central angle of measure θ radians is given by the product of the radius and the radian measure of the angle, or s = rθ,θ in radians. 8

9. Example of Finding Arc Length of a Circle • A circle has radius 18.2 cm. Find the length of the arc intercepted by a central angle having each of the following measures. • a) • b) 144 9

10. Example of Application • A rope is being wound around a drum with radius .8725 ft. How much rope will be wound around the drum it the drum is rotated through an angle of 39.72? • Convert 39.72 to radian measure. 10

11. Let’s Practice Another Application of Radian Measure Problem • Two gears are adjusted so that the smaller gear drives the larger one, as shown. If the smaller gear rotates through 225, through how many degrees will the larger gear rotate? 11

12. Let’s Practice Another Application of Radian Measure Problem (cont.)‏ • Find the radian measure of the angle and then find the arc length on the smaller gear that determines the motion of the larger gear. 12

13. Let’s Practice Another Application of Radian Measure Problem (cont.)‏ • An arc with this length on the larger gear corresponds to an angle measure θ, in radians where • Convert back to degrees. 13

14. How to Find Area of a Sector of a Circle? • A sector of a circle is a portion of the interior of a circle intercepted by a central angle. “A piece of pie.” • The area of a sector of a circle of radius rand central angle θis given by 14

15. Example • Find the area of a sector with radius 12.7 cm and angle θ = 74. • Convert 74 to radians. • Use the formula to find the area of the sector of a circle. 15

16. What is a Unit Circle? • A unit circle has its center at the origin and a radius of 1 unit. • Note: r = 1 • s = rθ, • s=θ in radians. 16

17. Circular Functions and their Reciprocals This is an example of a triangle in the 1st quadrant y 1 y θ y y x 17

18. Remember our two special triangles that make up the unit cirlce:

19. Let’s Look at the Unit Circle Again Because its made up of our “special” triangles. 19

20. Example of Finding Exact Circular Function Values • Find the exact values of • Evaluating a circular function at the real number is equivalent to evaluating it at radians. An angle of intersects the unit circle at the point . • Since sin θ = y, cos θ = x, and 20

21. IF AN ANGLE IN STANDARD POSITION MEASURES THE GIVEN RADIANS, DETERMINE WHICH QUADRANT IT’S TERMINAL SIDE LIES. II III I II

22. Change the given degree measure to radian measure in terms of π.

23. Change the given radian measure into degrees. =-57.3° =720° =33.75° =-140°

24. Find one positive and one negative angle that is coterminal with an angle measuring the given θ add 360° subtract 360° --290°, 430° add 2π subtract 2π add 360° subtract 360° --660°, 60° add 2π subtract 2π

25. Find the reference angle for the angle given: Is the acute angle formed with the x-axis 20° θ θ 20° one full revolution With left over θ θ -112.5°

26. Find the length of an arc that subtends an angle given, in a circle with diameter 20 cm. Write your answer to the nearest tenth 1.) 3.) 5.2 cm 15.7 cm 4.) 2.) 6.3 cm 10.5 cm

27. Find the degree measure of the central angle whose intercepted arc measures given, in a circle with radius 16 cm. 87 5.6 12 25 Now convert to degrees Now convert to degrees Now convert to degrees Now convert to degrees

28. Find the area, to the nearest tenth, of the sector of a circle defined by a central angle given in radians, and the radius given.

29. Find the values of the six trig functions of an angle in standard position if the point given lies on its terminal side. Use Pythagorean theorem to find the hypotenuse 5 (-1,5) (6,-8) (3,2) (-3,-4) θ -1 Use Pythagorean theorem to find the hypotenuse 6 θ -8 Use Pythagorean theorem to find the hypotenuse 2 θ 3 Use Pythagorean theorem to find the hypotenuse -3 θ -4

30. Suppose θ is an angle in standard position whose terminal side lies in the given quadrant. For each function, find the values of the remaining five trig functions of θ. Quadrant I Since we know cosine we can set up our triangle 4 θ 3 Then use Pythagorean theorem to find the other leg Quadrant IV Since we know sine we can set up our triangle θ Then use Pythagorean theorem to find the other leg -2

31. Determine if the following are positive, negative,zero, or undefined. Quadrant II Positive Not in Quad Undefinded Quadrant IV Negative Not in Quad Zero