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Understanding Trigonometric Functions for Angle Measurement

Trigonometric functions are essential in mathematics for measuring angles, with radian measure preferred over degrees. The conversion between degrees and radians is crucial for understanding angles in various contexts. This fundamental concept is key to exploring periodic phenomena and abstract mathematical spaces.

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Understanding Trigonometric Functions for Angle Measurement

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  1. Intro to Calculus March 2010

  2. The trigonometric functions are among the most fundamental in mathematics. They were initially developed to aid in the measurement of triangles and their angles, and they are used daily in surveying and navigation. However they can be used to describe any natural phenomenon that is periodic, and in higher mathematics they are fundamental tools for understanding many abstract spaces.

  3.  The primary use of trigonometric functions is in the measurement of angles. Although the ancient Babylonian degree unit of angle measure is still in wide use, in mathematics we prefer to use the radian measure. Given a circle centered at the origin in the Cartesian plane, imagine taking a radius and laying it along the outside circle, beginning at the x axis and going counterclockwise.

  4.  We begin with two basic measurements followed by two fundamental definitions.  Once around a circle is 360º.  And that the circumference of a circle is 2r

  5. 2

  6. Using the fact that 360º corresponds to 2Π radians, we can generate the following angle measures:  360º Dividing equals 180º which is radians 2 180º  Dividing gives us 90º which is radians 2 2

  7. Using the fact that 360º is 2 radians and 180º is radians, we can always convert degrees to radians by  Multiplying  the degree measurement by  º

  9. If we have a radian measure we can multiply by         180     180 180 60    So 60  1

  10. Degree measure of angle a circle and radian measure is based upon as another way to describe one complete circle. Degree measure of angle is based upon the in We can convert from radians to degrees And from degrees to radians.

  11. Convert to radians or degrees! Convert to radians or degrees! 34º 3π rad 6π rad  4  5 rad rad 732º   4π rad π rad 3 rad 46º 150º 2  2 rad rad 2010º  3

  12. Convert to radians 34º 15º 156º 272º 994º 52º 36º 174º 532º 732º 35º 37º 376º 631º 897º 4π rad 94º 324º 856º 1768º 74º 53º 163º 428º 2000º

  13. Convert to degrees 2π rad 7π rad 45π rad 2π rad 14π rad 9 rad 6π rad 6π rad 8π rad π rad 4π rad 73π rad 5π rad 3π rad 3π rad π rad 8 rad 25π rad 3π rad 4π rad

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