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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 abbreviation for tangent

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**Which of the following represents r in the figure below**

Which of the following represents r in the figure below? (Click on the blue.) x P(x,y) r y 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!

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**Which of the following represents sin in the figure below**

Which of the following represents sin in the figure below? (Click on the blue.) x P(x,y) r y 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.

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**Which of the following represents cos in the figure below**

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

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**Which of the following represents tan in the figure below**

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

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**In your notes, please copy this figure and the following three ratios:**

x P(x,y) r y

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**A few key points to write in your notebook: **

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 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.

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(–3,2) r –3 2 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

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**Example: If is a fourth-quadrant angle and sin = –5/13, find cos .**

Since is in quadrant IV, the coordinate signs will be (+x, –y), therefore x = +12.

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**Example: If is a second quadrant angle and cos = –7/25, find sin .**

Check Answer

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**Determine the signs of sin , cos , and tan according to quadrant**

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) r y r x y P(x,y) x P(–x, –y) r x y P(x, –y) r x y

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**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. y x All Sine Tangent Cosine

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**A handy pneumonic to help you remember! Write it in your notes!**

Students All x Take Calculus

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**Find the reference angle. Determine the sign by noting the quadrant.**

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 y P(x, –y) r x y x r P(–x, –y)

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**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

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**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

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**Try some sine problems now: Give each of the following in terms of the sine of a reference angle:**

Check Answer Check Answer Check Answer Check Answer

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**Can you complete this chart?**

60 30 45 60 30 45

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**Check your work!!!!!! Write this table in your notes!**

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**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:

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**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

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

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