7-4 Evaluating Trigonometric Functions of Any Angle

• Evaluate trigonometric functions of any angle

• Use reference angles to evaluate trigonometric functions

• Evaluate trigonometric functions of real numbers

222 ryx 22 yxr

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 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.

Recall

r

ysin

r

xcos

𝑡𝑎𝑛𝜃=𝑦𝑥,𝑥 ≠0

𝑐𝑠𝑐 𝜃=𝑟𝑦, y ≠0

𝑐𝑜𝑡 𝜃=𝑥𝑦, y≠0

s

In General

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 α.

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

:I I 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

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

Not all angles are SPECIAL. Sometimes you need to use your calculator. Be careful. Some problems are in degrees and some problems are in radians. Either switch back and forth between the two modes in your calculator. Or keep it in degree mode and convert quickly from radians to degrees first . . . sort of anyway.

Example: sin 217Make sure you are in DEGREE mode and just type it in your calculator!

The angle =217 is in Quadrant III; sin is negative in Quadrant III, so the sign of the angle makes sense.

𝑠𝑖𝑛217 °=− .602

Example: cos

If still in DEGREE mode, type the following:

If you switch to RADIAN mode, type the following:cos ¿

cos ¿

You try some: Give the value rounded to 3 places:

• sin 28

• cos 238

• tan 302

• cos (–15)

• sin /9

• cos (–2/5)

• tan 15/7

469.0Check Answer

530.0Check Answer

600.1Check Answer

966.0Check Answer

342.0Check Answer

309.0Check Answer

482.0Check Answer

Homework: Page 294-296, #5-7, 13-16, 17, 19, 21, 38, 40, 42, 44, 81-90