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Radian Measure and applications

Delve into the concept of radians in circular functions and trigonometry, discovering their importance in measuring angles and calculating areas of sectors and segments. Learn the fundamental relationships between radians and degrees. Test your knowledge with interactive exercises.

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Radian Measure and applications

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  1. Radian Measure and applications Chapter 2 Circular Functions and Trigonometry

  2. Measuring Angles • Why do we measure angles using degrees? • Why are there 360 of them in a circle? • What if I wanted to divide them into different pieces?

  3. A radian An Intro • Look at this diagram- applet:

  4. What is a radian? • The measure of the CENTRAL angle subtended by an arc EQUAL IN LENGTH to the radius of a circle. • Here is a radian. • (the green angle) • There are 2π radians in a circle(find out why later!) • Click HERE to see how you get a radian.. There are also some questions to test you/

  5. 1 Radian

  6. A radian • Definition: if the circumference of a circle is 2пr how many r’s will go around the circle?

  7. The relationship • Yes 2  is equal to one full turn of a circle • So 2  = 360 0 • Or it is easier to remember •  = 180 0

  8. Complete this table

  9. Common angles • Click here to see some common angles in radians

  10. Arc length • How do we find the arc length? • Length of an arc l = θ X circumference 2п • Since the circumference is 2пr • Then: • So: l = rθ • Remember the angle θ is in radians!

  11. Area of a sector • Area of sector A = θX Area of a circle 2п • Since the area of a circle is пr2 • Then: • So: A = ½ r 2θ • Please Remember to use RADIANS

  12. Arc length, area of a sector • Area of a sector proof: • Arc length: • Please remember these angles in these two formulae are all in RADIANS!

  13. Radian • Click here for a game • Match the angles in degrees with radians here • Radian practice with trig functions here

  14. Area of a segment • Remember another formula or remember the method using common sense? • Let’s use our common sense! • What steps would you have to take to find the area of the segment?

  15. Finding the Area of a Segment Please use Radians! Find the area of the sector using A = ½ r 2θ Find the area of the triangle using ½ absinC What is the area of the segment? Segment = Sector –Triangle

  16. How to find the segment? • Calculate the area of the segment in the following diagram. The radius is 10cm and the central angle is /3c • Calculate the area of the sector • Calculate the area of the triangle • Calculate thee area of the segment

  17. How do I find the area of a segment? • Look at the following diagram and follow the hint steps to find the area of the segment. • Shade in the segment in the circle below. Label the triangle AOB. Angle AOB = 0.5 radians and the radius is 12 cm. • Find the area of the sector • Find the area of the triangle using ½ absinC • What is the area of the segment?

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