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

222 ryx 22 yxr

x0

P(x,y)

r

y

x

y

Which of the following represents r in the figure below? (Click on the blue.)

2

xyr

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!

r

xsin 0 x,

x

ysin

x0

P(x,y)

r

y

x

y

Which of the following represents sin in the figure below? (Click on the blue.)

r

ysin

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.

r

xcos

r

ycos

x0

P(x,y)

r

y

x

y

Which of the following represents cos in the figure below? (Click on the blue.)

0 x,x

ycos

CORRECT! Yeah! Sorry. Wrong ratio.Oops! Try something else.

0 x,x

ytan

r

ytan

x0

P(x,y)

r

y

x

y

Which of the following represents tan in the figure below? (Click on the blue.)

r

xtan

CORRECT! Yeah! Try again.Try again.

x0

P(x,y)

r

y

x

y

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

r

ysin

r

xcos

0 x,x

ytan

222 ryx 22 yxr

0

P(x,y)

r

x

y

r

ysin

r

xcos

0 x,x

ytan

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.

Example: If the terminal ray of an angle in standard position passes through (–3, 2), find sin and cos .

13

132

13

13

13

2

13

2

r

ysin

13

133

13

13

13

3

13

3

r

xcos

1323

23

22

r)(r

yx

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

5

35

4

cos

sinCheck Answer

12

144

144

16925

135

2

2

222

x

x

x

x

)(x

13

5

13

5

r

y

r

ysin

13

12

r

xcos

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.

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

25

24sinCheck Answer

x0

P(–x,y)

r

y

0

P(–x, –y)

rx

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.

(neg)

(neg)

(pos)

II Quadrant

x

ytan

r

xcos

r

ysin

x

ytan

r

xcos

r

ysin

III Quadrant

x

ytan

r

xcos

r

ysin

I Quadrant

x

ytan

r

xcos

r

ysin

IV Quadrant

y

x

AllSine

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.

y

x

AllStudents

Take Calculus

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

x0

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.

180

180

2360

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.

4535180

180

:II quadrant in is 135 Since

3

Check Answer

Check Answer

30

Give each of the following in terms of the cosine of a reference angle:

Example: cos 160The 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

2cosCheck Answer

80cosCheck Answer

5cosCheck Answer

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

10sinCheck Answer

30sinCheck Answer

15sinCheck Answer

40sinCheck Answer

(degrees) (radians) sin cos

0 0 0 1

30 6

2

1

2

3

45

60

90 2

1 0

Can you complete this chart?

45

45

2

1

1

60

30

3

1

260

30

21

3

2

330

2

130

213

r

xcos

r

ysin

r,y,x

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

(degrees) (radians) sin cos

0 0 0 1

30 6

2

1

2

3

45 4

2

2

2

2

60 3

2

3

2

1

90 2

1 0

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:

(degrees) (radians) sin cos

0 0 0 1

30 6

2

1

2

3

45 4

2

2

2

2

60 3

2

3

2

1

90 2

1 0

2

245225 sinsin

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

2

2Check Answer

2

2Check Answer

2

2Check Answer

2

3Check Answer

2

3Check Answer

2

1Check Answer

2

2Check Answer

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