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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.

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Presentation on theme: "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."— Presentation transcript:

1 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

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

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

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

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

7 0 P(x,y) r x y 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 x 2 + y 2 = r 2 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 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 . (–3,2) r –3 2 Check Answer

9 Example: If  is a fourth-quadrant angle and sin  = –5/13, find cos . 13 –5 x 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 x 0 P(–x,y) r y 0 P(–x, –y) r x y P(x,y) 0 r x y 0 P(x, –y) r 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.

12 y x AllSine TangentCosine 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 y x AllStudents TakeCalculus A handy pneumonic to help you remember! Write it in your notes!

14 x 0 P(–x,y) r y 0 P(–x, –y) r x y P(x,y) 0 r x y 0 P(x, –y) r 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. 1.Find the reference angle. 2.Determine the sign by noting the quadrant. 3.Evaluate and apply the sign.

15 Example: Find the reference angle for  = 135 . 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

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

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

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 is negative. 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

22 Homework: Page , #1, 3, 11, 13, 15, 17


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