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7-3 Sine and Cosine (and Tangent) Functions 7-4 Evaluating Sine and Cosine

7-3 Sine and Cosine (and Tangent) Functions 7-4 Evaluating Sine and Cosine. sin is an abbreviation for sine cos is an abbreviation for cosine tan is an abbreviation for tangent. y. P(x,y ). r. y. 0. x. x.

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7-3 Sine and Cosine (and Tangent) Functions 7-4 Evaluating Sine and Cosine

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  1. 7-3 Sine and Cosine (and Tangent) Functions7-4 Evaluating Sine and Cosine sin is an abbreviation for sine cos is an abbreviation for cosine tan is an abbreviation for tangent

  2. y P(x,y) r y 0 x x Which of the following represents r in the figure below? (Click on the blue.) Close. The Pythagorean Theorem would be a good beginning but you will still need to “get r alone.” You’re kidding right? (xy)/2 represents the area of the triangle! CORRECT!

  3. y P(x,y) r y 0 x x Which of the following represents sin  in the figure below? (Click on the blue.) Sorry. Does Some Old Hippy Caught Another Hippy Tripping on Acid sound familiar? y/x represents tan. Sorry. Does SohCahToa ring a bell? x/r represents cos. CORRECT! Well done.

  4. y P(x,y) r y 0 x x Which of the following represents cos  in the figure below? (Click on the blue.) Oops! Try something else. Sorry. Wrong ratio. CORRECT! Yeah!

  5. y P(x,y) r y 0 x x Which of the following represents tan  in the figure below? (Click on the blue.) Try again. Try again. CORRECT! Yeah!

  6. y P(x,y) r y 0 x x In your notes, please copy this figure and the following three ratios:

  7. y P(x,y) r x 0 • A few key points to write in your notebook: • P(x,y) can lie in any quadrant. • Since the hypotenuse r, represents distance, the value of r is always positive. • The equation x2 + y2 = r2 represents the equation of a circle with its center at the origin and a radius of length r. • The trigonometric ratios still apply but you will need to pay attention to the +/– sign of each.

  8. (–3,2) r 2 –3 Example: If the terminal ray of an angle  in standard position passes through (–3, 2), find sin  and cos . You try this one in your notebook: If the terminal ray of an angle  in standard position passes through (–3, –4), find sin  and cos . Check Answer

  9. Example: If  is a fourth-quadrant angle and sin  = –5/13, find cos . x –5 13 Since  is in quadrant IV, the coordinate signs will be (+x, –y), therefore x = +12.

  10. Example: If  is a second quadrant angle and cos  = –7/25, find sin . Check Answer

  11. y y P(–x,y) r r x 0 0 y y x x 0 0 r r P(x, –y) P(–x, –y) Determine the signs of sin , cos  , and tan according to quadrant. Quadrant II is completed for you. Repeat the process for quadrants I, III, and IV. Hint: r is always positive; look at the red P coordinate to determine the sign of x and y. P(x,y) x

  12. y Sine All x Tangent Cosine • Check your answers according to the chart below: • All are positive in I. • Only sine is positive in II. • Only tangent is positive in III. • Only cosine is positive in IV.

  13. A handy pneumonic to help you remember! Write it in your notes! y Students All x Take Calculus

  14. y x 0 r P(x, –y) Let  be an angle in standard position. The reference angle  associated with  is the acute angle formed by the terminal side of  and the x-axis. y y P(–x,y) P(x,y) r r • Find the reference angle. • Determine the sign by noting the quadrant. • Evaluate and apply the sign. x x 0 0 y x 0 r P(–x, –y)

  15. Example: Find the reference angle for  = 135. Check Answer You try it: Find the reference angle for  = 5/3. You try it: Find the reference angle for  = 870. Check Answer

  16. Give each of the following in terms of the cosine of a reference angle: • Example: cos 160 • The angle =160 is in Quadrant II; cosine is negative in Quadrant II (refer back to All Students Take Calculus pneumonic). The reference angle in Quadrant II is as follows: =180 –  or =180 – 160 = 20. Therefore: cos 160 = –cos 20 • You try some: • cos 182 • cos (–100) • cos 365 Check Answer Check Answer Check Answer

  17. Try some sine problems now: Give each of the following in terms of the sine of a reference angle: • sin 170 • sin 330 • sin (–15) • sin 400 Check Answer Check Answer Check Answer Check Answer

  18. 30 60 45 60 30 Can you complete this chart? 45

  19. Check your work!!!!!! Write this table in your notes!

  20. Give the exact value in simplest radical form. Example:sin 225Determine the sign: This angle is in Quadrant III where sine isnegative. Find the reference angle for an angle in Quadrant III:  =  – 180 or  = 225 – 180 = 45. Therefore:

  21. You try some: Give the exact value in simplest radical form: • sin 45 • sin 135 • sin 225 • cos (–30) • cos 330 • sin 7/6 • cos /4 Check Answer Check Answer Check Answer Check Answer Check Answer Check Answer Check Answer

  22. Homework: Page 279-280, #1, 3, 11, 13, 15, 17

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