MATH 31 LESSONS

Chapters 6 & 7:

Trigonometry

1. Trigonometry Basics

Section 6.1: Functions of Related Values

Read Textbook pp. 250 - 267

A. Standard Position

Angles are created by rotating an “arm” from the

positive x-axis, which is called the initial arm.

Where the angle ends is called the terminal arm.

Counterclockwise angles are positive.

Clockwise angles are negative.

e.g.

Sketch the angle +200 in standard position.

y

x

y

xstart

Angles in standard position are measured from the positive x-axis, which is the initial arm.

y

xstart

It is useful to include the major angles at each axis. The clockwise direction is positive.

0, 360

90

180

270

y

xstart

Now, we sweep the arm 200 counterclockwise, since it is a positive angle.

0, 360

90

180

270

200

e.g.

Sketch the angle 310 in standard position.

y

x

y

xstart

Angles in standard position are measured from the positive x-axis, which is the initial arm.

It is useful to include the major angles at each axis. This time, we want the negative angles (clockwise).

y

xstart 0, 360

270

180

90

Now, we sweep the arm 310 clockwise, since it is a negative angle.

310

y

xstart 0, 360

270

180

90

Radians

Another measure of angle, apart from degrees, is radians.

To convert from radians to degrees (and vice versa),

use the following information:

2 radians = 360

or, radians = 180

y

x

These are the major angles at each axis in radians.

0, 2

2

2

3

Converting from Radians to Degrees

e.g. Convert radians to degrees.4

7

4

7 4

1807

300

Since radians = 180, you can simply substitute 180 wherever you see .

Converting from Radians to Degrees

e.g. Convert 252 to radians.

180

radians252

degrees goes on the bottom, since it must cancel

Recall, radians = 180

180

radians252

radians5

7

B. Trig Ratios

When we define the trigonometric ratios, we will use

a circle (rather than a triangle).

In this way, we can deal with angles that are

bigger than 180 (as well as negative angles).

Primary Trig Ratios

Consider a circle of radius r.y

x

r

We consider any point P

that is on the circumference

of the circle.

Its general coordinates

will be (x, y).

P (x, y)

We can create a triangle

with height y and base x.

The hypotenuse will be r,

since it represents the

radius of the circle.

r y

x

P (x, y)

Using the Pythagorean

theorem,

r y

x

P (x, y)

222 yxr

22 yxr r is the radius of the circle, so it must always be positive (r > 0)

We can now use

“Soh Cah Toa” to define

each primary trig ratio. r y

x

P (x, y)

“Soh Cah Toa”

r y

x

P (x, y)

""

""sin

hyp

opp

r

ysin

“Soh Cah Toa”

r y

x

P (x, y)

""

""cos

hyp

adj

r

xcos

“Soh Cah Toa”

r y

x

P (x, y)

""

""tan

adj

opp

x

ytan

Reciprocal Trig Ratios

sin

1csc

cos

1sec

tan

1cot

Ex. 1 Evaluate cos and csc if

Answer in exact values. Do not find the angle.

Try this example on your own first.Then, check out the solution.

2

,12

5tan

0

Determine the quadrant of the angle

2

y

x

2

x

The angle is in quadrant 2, where x is negative and y is positive

Sketch the triangle

Thus, x = -12 and y = 5

12

5tan

x

y Remember, x is negative and y is positive

12

5tan

r

x = -12

y = 5

Find r

r

x = -12

y = 5 222 yxr

22 yxr

22 512

13

Find cos

r = 13

x = -12

y = 5 r

xcos

13

12cos

Find csc

r = 13

x = -12

y = 5

sin

1csc

ry1

y

r

5

13

C. Reference and Coterminal Angles

Reference Angle

The reference angle is the acute angle (< 90)between the terminal arm and the nearest x-axis.

Reference angles are always positive.

e.g.

What is the reference angle for 260?

First, we sketch the angle.

y

xstart 0, 360

90

180

270

260

The reference angle is the angle between the terminal arm

and the nearest x-axis.

y

x18080

270

The reference angle is 80

Coterminal Angles

Two angles that have the same terminal arm

are called coterminal angles.

y

x

y

x

and are coterminal.

Note:

For any given reference angle (e.g. 50),there are an infinite number of coterminal angles.

y

x50

Note:

For any given reference angle (e.g. 50),there are an infinite number of coterminal angles.

y

x50

The smallest positive angle to the terminal arm (130)is called the principal angle.

y

x50

130

The next positive angle with the same terminal arm is constructed by adding 360 to the principal angle.

y

x50

130 + 360 = 490

The first negative angle with the same terminal arm is constructed by subtracting 360 from the principal angle.

y

x50

130 - 360 = -230

In general, we can find all coterminal angles by adding

or subtracting multiples of 360 from the principal angle.

i.e. If 1 and 2 are coterminal, then

2 = 1 + (360) n , where n I

n belongs to the integers.

Thus, n ... , -3, -2, -1, 0, 1, 2, 3, ...

If 1 and 2 are coterminal, then

2 = 1 + (360) n , where n I

or in radian form,

2 = 1 + 2 n , where n I

D. Exact Trig Values (Using Special Triangles)

It is crucial that you remember the

exact values for the trig ratios of the following angles:

30 45 60

To do so, we need to use special triangles.

60 "Soh Cah Toa"

60

32 2

1 1

3sin 60 = 2

cos 60 = 1 2

3tan 60 = 1

30 "Soh Cah Toa"

30

32 2

1 1

3cos 30 = 2

sin 30 = 1 2

3tan 30 = 1

45 "Soh Cah Toa"

45

2

1

1

2

sin 45 = 1

tan 45 = 1

2

cos 45 = 1

E. CAST

This is a simple but effective way to remember the signs

of all trig ratios in each quadrant.

C

AS

T

12

3 4

C

All +S

T

12

3 4

sin + cos + tan +

C

ASine +

T

12

3 4

sin + cos tan

C

AS

Tan +

12

3 4

sin cos tan +

Cos +

AS

T

12

3 4

sin cos + tan

F. Unit Circle

If we define the trig circle with a radius of 1 unit,

called the unit circle, then finding exact values for

the trig ratios is much more straightforward.

1

1

“Soh Cah Toa”

1 y = sin

x""

""sin

hyp

opp

yy

1

sin

The sine ratio is simply the y-coordinate.

“Soh Cah Toa”

1 y

x = cos ""

""cos

hyp

adj

xx

1

cos

The cosine ratio is simply the x-coordinate.

“Soh Cah Toa”

r y

x

P (x, y)

""

""tan

adj

opp

x

ytan

The tangent ratio remains y over x.

In general,

r y = sin

P (sin , cos )

x = cos

Building the unit circle ...

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

Start with the axes.

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

2

1,

2

330

6

2

2,

2

245

4

2

3,

2

160

3

Next, add the special triangle ratios in quadrant 1.

Remember, x = cos and y = sin

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

2

1,

2

330

6

2

2,

2

245

4

2

3,

2

160

3

It is crucial that you memorize this special triangle.

Ex. 2 Evaluate the following exactly (without your calculator):

Try this example on your own first.Then, check out the solution.

2tancos5

2sin3

Convert to degrees (if you need to)

2tancos5

2sin3

1802tan180cos52

180sin3

Recall, radians = 180

2tancos5

2sin3

1802tan180cos52

180sin3

360tan180cos590sin3

Evaluate the special angles

using the unit circle

(1, 0)

(0, 1)

(-1, 0)

(0, 1)

2

1,

2

330

6

2

2,

2

245

4

2

3,

2

160

3

190sin

Recall, sine is the y-coordinate

(1, 0)

(0, 1)

(-1, 0)

(0, 1)

90

1180cos

Recall, cosine is the x-coordinate

(1, 0)

(0, 1)

(-1, 0)

(0, 1)

180

01

0360tan

Recall, tangent is y / x

(1, 0)

(0, 1)

(-1, 0)

(0, 1)

360

Answer the question:

2tancos5

2sin3

360tan180cos590sin3

01513

8

Ex. 3 Evaluate the following exactly (without your calculator):

Try this example on your own first.Then, check out the solution.

3tan

6cos

4sin 2

Convert to degrees (if you need to)

Recall, radians = 180

3tan

6cos

4sin 2

3

180tan

6

180cos

4

180sin 2

3tan

6cos

4sin 2

3

180tan

6

180cos

4

180sin 2

60tan30cos45sin 2

Note that sin 2 = (sin ) 2

Evaluate the special angles

using the unit circle

(1, 0)

(0, 1)

(-1, 0)

(0, 1)

2

1,

2

330

6

2

2,

2

245

4

2

3,

2

160

3

These special angles can be read off the unit circle directly.

Answer the question:

3tan

6cos

4sin 2

60tan30cos45sin 2

21

23

2

3

2

22

1

3

2

3

4

2

21

23

2

3

2

22

2

3

2

1 1

1

3

2

3

4

2

21

23

2

3

2

22

Ex. 4 Express the following as a function of its related acute

angle and then evaluate:

Try this example on your own first.Then, check out the solution.

120sin

First, sketch the angle

y

xstart 0

90

180

120

Next, find the reference the angle

y

x60

120

The reference angle is 60

Using CAST, determine whether the trig ratio is

positive or negative

y

x60

Since the angle is in quadrant 2, sine is positive.

AS

T C

Express the trig ratio in terms of the reference angle:

120sin

60sin

Use the unit circle

to evaluate the special

angle exactly:

120sin

60sin

(1, 0)

(0, 1)

(-1, 0)

(0, 1)

2

3,

2

160

3

2

3

Recall, sine is the y-coordinate.

Ex. 5 Express the following as a function of its related acute

angle and then evaluate:

Try this example on your own first.Then, check out the solution.

6

11cot

First, convert to degrees and a primary trig ratio:

6

11cot

6

18011cot

330cot

Recall, radians = 180

6

11cot

6

18011cot

330cot

330tan

1

Next, sketch the angle

y

xstart 0, 360

90

180330

270

Next, find the reference the angle

The reference angle is 30

y

x30

330

Using CAST, determine whether the trig ratio is

positive or negative

Since the angle is in quadrant 4, tangent is negative (only cosine is positive)

C

y

x30

AS

T

Express the trig ratio in terms of the reference angle:

6

11cot

330tan

1

30tan

1

Use the unit circle

to evaluate the special

angle exactly:

Recall, tangent is y / x.

(1, 0)

(0, 1)

(-1, 0)

(0, 1)

2

1,

2

330

6

232

130tan

232

130tan

3

1

Answer the question:

6

11cot

30tan

1

311

3