1 A unit circle has its center at the origin and a radius of 1 unit. 3.3 Definition III: Circular Functions

2 Circular Functions

3 Unit Circle

4 Domains of the Circular Functions Assume that n is any integer and s is a real number. Sine and Cosine Functions: ( ,  ) Tangent and Secant Functions: Cotangent and Cosecant Functions:

5 Evaluating a Circular Function Circular function values of real numbers are obtained in the same manner as trigonometric function values of angles measured in radians. This applies both to methods of finding exact values (such as reference angle analysis) and to calculator approximations. Calculators must be in radian mode when finding circular function values.

6 Example: 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 s = y, cos s = x, and

7 Example: Approximating Find a calculator approximation to four decimal places for each circular function. (Make sure the calculator is in radian mode.) a) cos 2.01  .4252b) cos.6207 .8135 –For the cotangent, secant, and cosecant functions values, we must use the appropriate reciprocal functions. c) cot

8 The length s of the arc intercepted on a circle of radius r by a central angle of measure  radians is given by the product of the radius and the radian measure of the angle, or s = r ,  in radians. 3.4 Arc Length and Area of a Sector

9 Example: Finding Arc Length 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 

10 Example: Finding Arc Length -- continued a) r = 18.2 cm and  = b) convert 144  to radians

11 Example: Finding a Length 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  ? Convert to radian measure.

12 Example: Finding an Angle Measure 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?

13 Solution 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.

14 Solution continued An arc with this length on the larger gear corresponds to an angle measure , in radians where Convert back to degrees.

15 Area of a Sector 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 r and central angle  is given by

16 Example: Area Find the area of a sector with radius 12.7 cm and angle  = 74 . Convert 74  to radians. Use the formula to find the area of the sector of a circle.