Trigonometric Functions of Any Angles

Objective:

To evaluate trig functions of any angle by using reference angles

θ(x, y)

r

2 2r = x + y

sin θ =yr

cos θ =xr

tan θ =yx

csc θ =ry

sec θ =rx

cot θ =xy

Evaluate the six trigonometric functions if the point (-4,3) lies on the terminal side of an angle θ.

2 2r = -4 + 3

r = 16 + 9

r = 25

r = 5

sin θ =35

cos θ =-45

tan θ =3-4

csc θ =53

sec θ =5-4

cot θ =-43

Evaluate the six trigonometric functions if the point (1,-3) lies on the terminal side of an angle θ.

2 2r = 1 + -3

r = 1 + 9

r = 10

sin θ =

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

-3

10-3 10

10

1

101010

-31

-3

10-3

101

1-3

Evaluate the six trigonometric functions if the point (-7,-4) lies on the terminal side of an angle θ.

2 2r = -7 + -4

r = 49+16

sin θ =

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

-4

65-4 65

65

-7

65-7 65

65

-4-7

47

65-4

65-7

74

r = 65

Evaluate the six trigonometric functions if the point (3, 0) lies on the terminal side of an angle θ.

2 2r = 3 + 0

r = 9

r = 3

sin θ =

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

03

0

33

1

03

0

30

= undefined

130

= undefined

When Trig Functions are +/-

(+,+)(-,+)

(-,-) (+,-)

sin, csc +

cos, sec –

tan, cot –

sin, csc –

cos, sec –

tan, cot +

sin, csc –

cos, sec +

tan, cot –

sin, csc +

cos, sec +

tan, cot +III

III IV

Name the quadrant in which the angle lies.

sinθ < 0 and cosθ < 0

sinθ > 0 and tanθ < 0

III

II

Evaluate the six trigonometric functions if the point (0, 3) lies on the terminal side of an angle θ.

2 2r = 0 + 3

r = 9

r = 3

sin θ =

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

03

0

33

1

03

30

= undefined

1

30

= undefined 0

Evaluate the six trigonometric functions if the point (-3, 0) lies on the terminal side of an angle θ.

2 2r = -3 + 0

r = 9

r = 3

sin θ =

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

03

0

-33

1

0-3

0

30

= undefined

-1-30

= undefined

Evaluate the six trigonometric functions if the point (0, -3) lies on the terminal side of an angle θ.

2 2r = 0 + -3

r = 9

r = 3

sin θ =

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

-1

03

0

0-3

0

-33

= -1

30

-30

= undefined

= undefined

(3,0)

If (3,0) is on the terminal side of the angle, then θ = 0°.

(0,3)

If (0,3) is on the terminal side of the angle, then θ = 90°

90°

(-3,0)

If (-3,0) is on the terminal side of the angle, then θ = 180°

(0,-3)

If (0,-3) is on the terminal side of the angle, then θ = 270°

180°270°

Quadrantal Angle

• Angle whose terminal side falls on an axis.

• Examples: 0°, 90°, 180°, 270°, 360°

Trig Functions of Quadrantal Angles

θ sin cos tan csc sec cot

0°

90°

180°

270°

360°

0 1 0 ud. 1 ud.

1 0 ud. 1 ud. 0

0 -1 0 ud. -1 ud.

-1 0 ud. -1 ud. 0

0 1 0 ud. 1 ud.

Reference Angles

• The acute angle formed by the terminal side of the angle and the horizontal axis (x-axis).

Remember:Must be positiveMust be acute

Find the reference angle.

140°

180° – 140° = 40°

Find the reference angle.

230°

230° – 180°

= 50°

Short-Cuts for Reference Angles

Quadrant θ in degrees θ in radians

I

II

III

IV

If θ goes around more than once…

θ θ

180° - θ π - θ

θ - 180° θ - π

360° - θ 2π - θθ - 360°.

Then proceed with above.

θ - 2π

Then proceed with above.

Find the reference angle.

310

170

305

360° - 310° = 50°

180° - 170 ° = 10°

360° - 305° = 55°

Find the reference angle.

3

2

5

7

4

2

3

2

1 3

3 2

3 3

3

5

72

4

2 7

1 4

8 7

4 4

4

Evaluate the following.

cos 225

tan120

225° - 180° = 45°

cos(45°)22

180° - 120° = 60°

tan(60°) 3

22

3

Quadrant III

Quadrant II

In Quadrant III, cos is negative.

In Quadrant II, tan is negative.

Evaluate the following.

tan 225 225 180 45

tan 45 1Quadrant III

1

In Quadrant III, tan is positive.

Evaluate the following.

sin 210

210 180 30

sin 30 1

2

Quadrant III1

2

In Quadrant III, sin is negative.

= 55°

= 70°

Find the reference angle.

375

470

595

375° - 360° = 15°

470° - 360 ° = 110°

595° - 360° = 235°

180° - 110 °

235° - 180 °

Evaluate the following.

sin 390

390° - 360° = 30°

sin(30°)12

It’s larger than 360°!

Quadrant I.

12

In Quadrant I, sin is positive.

Evaluate the following.

9tan

4

It’s larger than 2π!

92

4

9 2

4 1

9 8

4 4

4

tan4

1

Quadrant I!1

If θ is negative…

1. Add 360° (or 2π) until you get a positive angle.

2. Proceed as usual.

= 68°

Find the reference angle.

-275

-190

-112

-275° + 360°= 85°

-190° + 360 °= 170°

-112° + 360°= 248°

248° - 180 °

180° - 170° = 10°

Find the reference angle.

3

4

32

4

3 2

4 1

3 8

4 4

5

4

Quadrant III.5

4

5

4 1

5 4

4 4

4

Evaluate the following.

sin( 120 )

-120 + 360 = 240°It’s negative.

Quadrant III.

240° - 180° = 60°

sin(60°)32

32

In Quadrant III, sin is negative.