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Evaluating Trig Functions Of Any Angle TUTORIAL

Evaluating Trig Functions Of Any Angle TUTORIAL. Click the speaker icon on each slide to hear the narration. First Concept: Evaluating a trig function of a special angle. Sketch the angle in standard position Determine the reference angle

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Evaluating Trig Functions Of Any Angle TUTORIAL

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  1. Evaluating Trig Functions Of Any AngleTUTORIAL Click the speaker icon on eachslideto hear the narration

  2. First Concept: Evaluating a trig function of a special angle • Sketch the angle in standard position • Determine the reference angle • Draw the triangle showing x, y, r with their values, based on the side ratios of the reference angle • Take the appropriate ratio of the sides • Simplify your ratio, if necessary

  3. Example: • Sketch the terminal side of the angle in standard position

  4. Example: • Find the reference angle:

  5. Example: • Label the values of x, y, and r, paying close attention to the signs (r is always positive): 1 60 2

  6. Example: • Compute the tangent ratio: 1 60 2

  7. Second Concept: Trig functions of angles that lie on the axes • Trig functions of 90, 180, 270, and 360 can be tricky • Steps: • Draw the angle and indicate x, y, and r • Use the following definitions:

  8. Example: Functions of 270 • Draw the angle in standard position 270

  9. Example: Functions of 270 • Indicate the coordinates of the endpoint of the terminal ray (always make r = 1) x = 0 y = –1 r = 1

  10. Example: Functions of 270 • Take the appropriate ratios to compute sin, cos, and tan x = 0 y = –1 r = 1

  11. Third Concept: Angles with similar ratios • Every angle with the same reference angle will have a similar ratio • Identical to each other, or… • Different sign from each other • Use knowledge of the quadrants and x, y, r to know whether the ratio is positive or negative in that quadrant • r is always positive

  12. Signs of functions in each quadrant • sin = y/r, so sin is positive where y is positive (Quadrants 1 and 2) and negative where y is negative (Quadrants 3 and 4) • cos = x/r, so cos is positive where x is positive (Quadrants 1 and 4) and negative where x is negative (Quadrants 2 and 3) • tan = y/x, so tan is positive where x and y have the same sign (Quadrants 1 and 3) and negative where x and y have different signs (Quadrants 2 and 4)

  13. Summary chart Sin is positive, others are negative All trig functions are positive x is negy is pos x and y are positive Tan is positive, others are negative Cos is positive, others are negative Mnemonic: x and y are negative x is posy is neg S A T C “All Students Take Calculus”“All Schools Torture Children” “Avoid Silly Trig Classes”

  14. Example: cos 35 = 0.819 • Other angles in the family (meaning they have a reference angle equal to 35) • In the second quadrant, 145 has the same reference angle and the cosine is negative, socos 145 = –0.819 • In the third quadrant, 215 has the same reference angle and the cosine is negative, socos 215 = –0.819 • In the fourth quadrant, 325 has the same reference angle and the cosine is positive, so cos 325 = 0.819

  15. The End • Hope you enjoyed the show!

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