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Chapter 3

Chapter 3. Radian Measure and the Unit Circle Approach. Chapter 3 Overview. Chapter 3 Objectives. Convert between degrees and radians. Calculate arc length and the area of a circular sector. Relate angular and linear speeds.

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Chapter 3

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  1. Chapter 3 Radian Measure and the Unit Circle Approach

  2. Chapter 3Overview

  3. Chapter 3Objectives • Convert between degrees and radians. • Calculate arc length and the area of a circular sector. • Relate angular and linear speeds. • Draw the unit circle and label the sine and cosine values for special angles (in both degree and radians.)

  4. Skills Objectives Calculate the radian measure of an angle. Convert between degrees and radians. Calculate trigonometric function values for angles given in radians. Conceptual Objectives Understand that degrees and radians are both measures of angles. Realize that radian measure allows us to write trigonometric functions as functions of real numbers. Section 3.1Radian Measure

  5. Radian Measure To correctly calculate radians from the formula , the radius and arc length must be expressed in the same units.

  6. Your Turn: Calculate Radian Measure • What is the measure (in radians) of a central angle  that intercepts an arc of length 12 mm on a circle with radius 4 cm? • Solution: First, convert 4 cm to 40 mm. • Next, calculate  =  0.3.

  7. Converting Between Degrees and Radians

  8. Your Turn : Converting Between Degrees and Radians • Convert 60° to radians. • Solution: r = 60 = or 1.047. • Convert to radians. • Solution: d = = 270°.

  9. Special Angles in the Unit Circle We can now draw the unit circle with special angles in degrees and radians.

  10. Skills Objectives Calculate the length of an arc along a circle. Find the area of a circular sector. Solve application problems involving circular arc lengths and sectors. Conceptual Objectives Understand that to use the arc length formula, the angle measure must be in radians. Section 3.2 Arc Length and Area of a Circular Section

  11. Arc Length

  12. Your Turn: Calculating Arc Length • In a circle with radius 15 inches, an arc is intercepted by a central angle with measure . Find the arc length. • Solution: s = 15 = 5p inches. • In a circle with radius 20 m, an arc is intercepted by a central angle with measure 113°. Find the arc length. Approximate the arc length to the nearest meter. • Solution: s = ≈ 39 m

  13. Calculating Area of a Circular Sector • Find the area of a slice of pizza (cut into 8 equal pieces) if the entire pizza has a 16-inch diameter. • Solution: r = 8, r = , so area =  25 in.2 .

  14. Skills Objectives Calculate linear speed. Calculate angular speed. Solve application problems involving angular and linear speeds. Conceptual Objectives Relate angular speed to linear speed. Section 3.3 Linear and Angular Speeds

  15. Linear and Angular Speed

  16. Your Turn: Linear Speed • A car travels ata constant speed around a circular track with circumference equal to 3 miles. If the car records a time of 12 minutes for 7 laps, what is the linear speed in miles per hour? • Solution: t = 12 minutes, or hour and s = 7 X 3. • V = = = 105 mph.

  17. Relationship Between Linear and Angular Speeds

  18. Skills Objectives Draw the unit circle illustrating the special angles and label the sine and cosine values. Determine the domain and range of trigonometric (circular) functions. Classify circular functions as even or odd. Conceptual Objectives Understand that trigonometric functions using the unit circle approach are consistent with both of the previous definitions (right triangle trigonometry and trigonometric functions of nonacute angles in the Cartesian plane). Relate x-coordinates and y-coordinates of points on the unit circle to the values of cosine and sine functions. Visualize periodic properties of trigonometric (circular) functions. Section 3.4 Definition 3 of Trigonometric Functions: Unit Circle Approach

  19. The Unit Circle

  20. Common Mistake: Using the Wrong Mode on a Calculator Many calculators automatically reset to degree mode after every calculation, so be sure to always check what mode the calculator indicates.

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