Trig Functions of Any Angle

Section 4.4

In first section, we calculated trig functions for acute angles.

In this section, we are going to extend these basic definitions to cover any angle.

θ

θ

Plot the point (-3,4)

Label the hypotenuse r and find its length.

r43

22 r = 5

5

-3

4θ Sin θ =

Cos θ =

Tan θ =

54

53

34

Definitions of Trig Functions of Any AngleLet θ be an angle in standard position with

(x,y) a point on the terminal side. Then:

ry

rx

xy

Csc θ =

Sec θ =

Cot θ =

Sin θ =

Cos θ =

Tan θ =

yr

xr

yx

Find the 6 trig functions of θ given that the ray ends at the point (-15, -8)

-15

-8 17

178

1715

158

Csc θ =

Sec θ =

Cot θ =

Sin θ =

Cos θ =

Tan θ =

817

1517

815

Find the 6 trig functions of θ given that the ray ends at the point (12, -5)

12

-5 13

135

1312

125

Csc θ =

Sec θ =

Cot θ =

Sin θ =

Cos θ =

Tan θ =

513

1213

512

QuadrantsIn which quadrants was the Sine positive?

I and II

In which quadrants was the Cosine positive?I and IV

In which quadrants was the Tangent positive?I and III

Quadrants

All Trig Functionsare positive

Sine is positive

Cosine is positive

Tangent is positive

AllStudents

Take Calculus

What quadrant is θ in if:

a) Sin θ > 0 and Cos θ < 0

b) Tan θ > 0 and Cos θ < 0

c) Sin θ < 0 and Tan θ < 0

d) Cos θ > 0 and Tan θ > 0

→ II

→ III

→ IV

→ I

Given that Tan θ = - and Sin θ > 0, find the

remaining 5 trig functions of θ.24

7

What quadrant? II

-24

7 2525

7

2524

247

Csc θ =

Sec θ =

Cot θ =

Sin θ =

Cos θ =

Tan θ =

725

2425

724

Given that Cos θ = - and Sin θ < 0, find the

remaining 5 trig functions of θ.5

4

What quadrant? III

-4

5-3

53

54

43

Csc θ =

Sec θ =

Cot θ =

Sin θ =

Cos θ =

Tan θ =

35

45

34

Given that Sin θ = - and Tan θ < 0, find the

remaining 5 trig functions of θ.17

15

What quadrant? IV

-1517

817

15

178

815

Csc θ =

Sec θ =

Cot θ =

Sin θ =

Cos θ =

Tan θ =

1517

817

158

What did we learnHow to find the trig functions of an angle

given a point on its terminal side

How to determine the quadrant of an angle based on trig functions

How to find the trig functions based on one function and criteria

Homework: Page 297, 1-24 odd

Find the Sin, Cos, and Tan trig functions of θ given that the ray ends at the point (5,0)

5

y = 005

0

155

050

Sin θ =

Cos θ =

Tan θ =

Quadrant AnglesOn our Cartesian plane, we

have 5 critical points:

2

3

2

0

2

Find the Sine of these 5 angles

Sin 0 = 0

Sin = 1 2

Sin π = 0

Sin = -1 2

3

Sin 2π = 0

Graph of the Sine CurveUsing these 5 points, we can create the Sine

Curve

20

2

2

3

Quadrant AnglesUsing the same process, find the Cos of the 5

critical points.

Cos 0 = 1

Cos = 0 2

Cos π = -1

Cos = 0 2

3

Cos 2π = 1

Graph of the Cosine CurveUsing these 5 points, we can create the Sine

Curve

20

2

2

3

Reference AnglesThe acute angle formed by the terminal side

of an angle and the horizontal axis.

For an angle θ, we use θ’ to denote the reference angle

Reference AnglesWhat is the reference angle for

210º

Where is there an acute angle

between the terminal side of the

angle and the horizontal axis?

θ’ = 210 – 180 = 30º

Reference AnglesFind the reference angles for the following:

a) 330º

b) 225º

c) -225º

d) 750º

= 360º - 330º = 30º

= 225º - 180º = 45º

= -180º - -225º = 45º

= 750º - 720º = 30º

Reference AnglesIn general, for any angle θ

θ’ = θθ’ = 180 - θθ’ = π - θ

θ’ = θ - 180θ’ = θ - π

θ’ = 360 - θθ’ = 2π - θ

Reference AnglesFind the reference angle for

2nd Quadrant: → π – θ

= π –

=

4

3

4

3

4

Reference AnglesSo far, all we have been finding are reference

angles.

We use reference angles to find the exact value of angles that are not acute.

We will use this for the remainder of the year.

“GTK” – Good to Know

Finding the Exact Value1. Find the reference angle

2. Find the trig function of the reference angle

3. Check the sign of the function

Sin 200º1. Find the reference angle

2. Find the Sin of the reference angle

3. Is it positive or negative?

Cos 330º1. Find the reference angle

2. Find the Sin of the reference angle

3. Is it positive or negative?

Find the Sin, Cos, and Tan of 135ºReference Angle =

Quadrant =

Sin 135º =

Cos 135º =

Tan 135º =

Find the Sin, Cos, and Tan of -240ºReference Angle =

Quadrant =

Sin -240º =

Cos -240º =

Tan -240º =

Find the Sin, Cos, and Tan ofReference Angle =

Quadrant =

Sin =

Cos =

Tan =

4

7

4

7

4

7

4

7

Find the:a) Sin

b) Csc

c) Tan

d) Csc

e) Cot

4

5

3

2

6

11

6

7

3

4