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Exploring Trigonometric Ratios and Functions: Angles, Radians, and Sectors

In this chapter, we focus on the fundamentals of trigonometric ratios and functions, including understanding angles in standard position and radian measures. Learn how to identify positive and negative angles, determine coterminal angles, and convert between degrees and radians. We also explore sectors of circles, defining the arc length and area formulas needed to solve for these quantities. Engage with examples and check your understanding through practice problems.

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Exploring Trigonometric Ratios and Functions: Angles, Radians, and Sectors

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Presentation Transcript

  1. Math III Accelerated Chapter 13 Trigonometric Ratios and Functions

  2. Warm Up 13.2 • Write the expression in simplest form.

  3. 13.2 Define General Angles and Use Radian Measure • Objective: • Use general angles that may be measured in radians.

  4. Angles in Standard Position • In the coordinate plane, an angle can be formed by fixing one ray, called the _______________ side, and rotating the other ray, called the ____________________ side, about the ____________________. • An angle is in standard position if its vertex is at ___________ and its initial side lies on the positive ____________________ .

  5. Positive and Negative Angles • A positive angle opens _________________. • A negative angle opens _________________.

  6. Example 1 • Draw a 405˚ angle in standard position. • Draw a –65˚ angle in standard position.

  7. Coterminal Angles • Angles that share the same terminal side are called coterminal angles. • Coterminal angles can be found by adding (or subtracting) multiples of 360˚ to (from) the given angle.

  8. Example 2 • Find one positive and one negative angle that are coterminal with 210˚. Sketch your results.

  9. Checkpoints 1 & 2 • Draw an angle with the given measure in standard position. Find one positive and one negative coterminal angle. • 485° • –75°

  10. Radian Measure • Angles can be measured either in degrees or in radians.

  11. Converting Between Degrees and Radians • Degrees to Radians Radians = Degrees • • Radians to Degrees Degrees = Radians •

  12. Example 3 • Convert : • 315˚ to radians • π/6 radians to degrees

  13. Checkpoints 3 & 4 • Convert the degree measure to radians or the radian measure to degrees. • 200˚ • π/5

  14. Sectors of Circles • A sector is a region of a circle that is bounded by two radii and an arc of the circle. • The central angle θof a sector is the angle formed by the two radii. r θ r

  15. Arc Length and Area of a Sector • The arc length s and areaA of a sector with radiusr and central angleθ (measured in radians) are: • Arc Length s = r θ • AreaA = ½ r2 θ s θ r

  16. Example 4 • Find the arc length and area of a sector with a radius of 15 inches and a central angle of 60˚.

  17. Checkpoint 5 • Find the arc length and area of a sector with a radius of 5 feet and a central angle of 75˚.

  18. Homework 13.2 • Practice 13.2

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