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1 6.1 Radian Measure

14.1 Angles geometry review and more!. 1 6.1 Radian Measure. Finally, what that other mode on our calculator is all about!. OBJECTIVES. Students will understand the meaning of radian measure Students will convert between degrees and radians

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1 6.1 Radian Measure

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  1. 14.1 Angles geometry review and more! 16.1 Radian Measure Finally, what that other mode on our calculator is all about!

  2. OBJECTIVES • Students will understand the meaning of radian measure • Students will convert between degrees and radians • Students will find function values for angles in radians

  3. Geometry Review • -angles can be named with three or one letters • Complementary angles add up to 90. • Supplementary angles add up to 180. • Angle 1 and 2 are supplementary. Angle 1 measures 2x + 6, Angle 2 measures 4x. Find x and the measure of the angles.

  4. The measure of an angle is determined by rotating a ray starting at one side of the angle called the initial side to the other side called the terminal side. • A counterclockwise rotation generates a positive measure, a clockwise rotation generates a negative measure. • An angle is said to lie in the quadrant in which its terminal side lies. (ex. An acute angle is in quadrant 1.) -We will be using the Greek letter theta θ to name angles.

  5. Calculating with degrees, minutes and seconds • One minute (1’) is equal to 1/60 degrees, or there are 60 minutes in one degree • One second (1”) is equal to 1/60 minutes and 1/3600 degrees. There are 60 seconds in a minute. • 12º42’38” represents 12 degrees 42 minutes 38 seconds • Ex. Add : 51º29’ + 32º46’

  6. Converting between decimal degrees and Degree, Minutes and Seconds • Convert 74º8’14” to decimal degrees to the nearest thousandth. • Or use your calculator!

  7. Quadrantal Angles • Angles are in standard position if the vertex is at the origin and the initial side lies on the positive x axis. • Angles in standard position whose terminal sides lie on the x or y axis are quadrantal angles. Ex. 90, 180 or 270 degrees or any multiple of these.

  8. Coterminal Angles. • The measures of coterminal angles differ by 360º • Coterminal angles have the same initial side and terminal side but have different amounts of rotation. Ex. 60º and 420º are coterminal angles. Can you name another pair of coterminal angles?

  9. Finding measures of coterminal angles. • Find angles of least possible positive measure coterminal with each angle. • A) 908º B.) -75º C.) -800º 188, 285, 280

  10. What is a radian? • Definition: An angle with its vertex at the center of a circle that intercepts an arc on the circle equal to the radius of the circle has a measure of 1 radian. • Θ = 1 radian • So 2 radians would measure an angle with an arc equal to twice the radius

  11. Converting between degrees and radians • To convert from DEGREES to RADIANS: • Multiply a degree measure by π/180 radians and then simplify. To convert from RADIANS to DEGREES: Multiply a radian measure by 180˚/π and then simplify. Example: 45˚ to radians…45(π/180) = 45π/180=π/4 radians (Notice I left it in fraction form) Example: 9π/4 to degrees…9π/4(180/π)=9(180)/4=405˚

  12. You try • Convert 108˚ to radians • 3π/5 radians (leave in fraction!) • Convert 11π/12 to degrees • 165˚ • You can use the angle key on your calculator, you must hit the 2nd key then the angle key. • We can have negative radian measures, just like we can have negative degree measures! A counterclockwise rotation generates a positive measure, a clockwise rotation generates a negative measure.

  13. NOTE: • If there is no unit that is specified in a problem or a diagram, then it is understood that the angle is measured in radians. • So, only work in degrees from now on if the problem specifically says to. • Don’t forget to use your mode key!

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