1

Chapter 4 Review of the Trigonometric Functions

2

Standard Position

Vertex at origin

The initial side of an angle in standard position is always located on the positive x-axis.

3

Positive and negative angles

When sketching angles, always use an arrow to show direction.

4

Angles in standard position are often classified according to

the quadrant in which their terminal sides lie.

Example:

50º is a 1st quadrant angle.

208º is a 3rd quadrant angle. II I

-75º is a 4th quadrant angle. III IV

Classifying Angles

5

Classifying Angles

Standard position angles that have their terminal side on

one of the axes are called ______________ angles.

For example, 0º, 90º, 180º, 270º, 360º, … are quadrantal angles.

6

1 minute (1') = degree (°) OR 1° = ______ '

1 second (1") = _____ minute (') OR 1' = _______"

Therefore, 1 second (1") = ________ degree (°)

Example

Convert to decimal degrees (to three decimal places): 52 15'42"

Degrees, minutes, and seconds

7

Conversions between decimal degrees and degrees, minutes, seconds can be easily accomplished using your TI graphing calculator.

The ANGLE menu on your calculator has built-in features for converting between decimal degrees and DMS.

Degrees, minutes, and seconds

Note that the seconds () symbol is not in the ANGLE menu.

Use for symbol.

8

Using your TI graphing calculator,

1) Convert to decimal degrees to the nearest hundredth of a degree.

2) Convert 57.328° to an equivalent angle expressed to the nearest second.

14 32 '18"

PracticeNOTE: SET MODE TO DEGREE

9

Coterminal Angles

Angles that have the same initial and terminal sides

are coterminal.

Angles and are coterminal.

10

Examples of Coterminal Angles

Find one positive and one negative coterminal angle for each angle given.

a) 125 b) 240 34' c) 311.8

11

Take a look at the right triangle, with an acute angle, , in the figure below.

Notice how the three sides are labeled in reference to .

The sides of a right triangle

Side adjacent to

S

ide

op

po

site

Hypotenuse

12

To remember the definitions of Sine, Cosine and Tangent, we use the acronym :

“SOH CAH TOA”

Definitions of the Six Trigonometric Functions

O A O

H HS C

AT

13

Definitions of Trigonometric Functions of an Angle

Let be an angle in standard position with (x, y) a point on the terminal side of and r is the distance from the origin to the point. Using the Pythagorean theorem, we have .

Definitions of the Trig Functions

2 2r x y

sin csc

cos sec

tan cot

y r

r y

x r

r xy x

x y

y

x

(x, y) r

14

Let (12, 5) be a point on the terminal side of . Find the value of the six trig functions of .

Example

y

r5

x

(12, 5)

12

First you must find the value of r:

2 2r x y

15

Example (cont)sin

cos

tan

csc

sec

cot

y

rx

ry

xr

y

r

xx

y

r5

x

(12, 5)

12

16

Given that is an acute angle and , find the exact value of the five remaining trig functions of .

Example8

cos17

17

Find the value of tan given csc = 1.02, where is an acute

angle. Give answer to three significant digits.

Example

18

The 45º- 45º- 90º Triangle

Special Right Triangles

1

12

45º

45º

Find the exact values of the six trig functions for 45

sin 45 = csc 45 =

cos 45 = sec 45 =

tan 45 = cot 45 =

Ratio of the sides:

19

The 30º- 60º- 90º Triangle

Special Right Triangles

Find the exact values of the six trig functions for 30

sin 30 = csc 30 =

cos 30 = sec 30 =

tan 30 = cot 30 =

1

3

60º

30º

2

Ratio of the sides:

20

The 30º- 60º- 90º Triangle

Special Right Triangles

Find the exact values of the six trig functions for 60

sin 60 = csc 60 =

cos 60= sec 60 =

tan 60 = cot 60 =1

3

60º

30º

2

Ratio of the sides:

21

Make sure the MODE is set to the correct unit of angle measure (i.e. Degree vs. Radian)

Example:

Find to three significant digits.

Using the calculator to evaluate trig functions

cos 37.8

22

For reciprocal functions, you may use the reciprocal button , but DO NOT USE THE INVERSE FUNCTIONS (e.g. )!

Example:

1. Find 2. Find

(to 3 significant dig) (to 4 significant dig)

Using the calculator to evaluate trig functions

csc84.1 cot 57 14 ' 38"

23

Angles and Accuracy of Trigonometric Functions

Measurement of Angle to Nearest

Accuracy of Trig Function

1° 2 significant digits

0. 1° or 10’ 3 significant digits

0. 01° or 1’4 significant digits

24

The inverse trig functions give the measure of the angle if we know the value of the function.

Notation:The inverse sine function is denoted as sin-1x or arcsinx. It means “the angle whose sine is x”.

The inverse cosine function is denoted as cos-1x or arccosx. It means “the angle whose cosine is x”.

The inverse tangent function is denoted as tan-1x or arctanx. It means “the angle whose tangent is x”.

25

Example

1 1sin2

is the angle whose sine is 1

2

Think of the related statement

must be 30°, therefore

1sin

2

1 1sin 302

26

Examples

Given that 0°≤ ≤ 90°, use an inverse trig functions to find the

value of in degrees.

31. tan 2. sin 0.25

5

To nearest 0.1 To 2 sig. dig.

27

Examples

Given that 0°≤ ≤ 90°, use an inverse trig functions to find the

value of in degrees.

3. cos 1 4. sin 3.22

28

Example

Solve the right triangle with the indicated measures.

1. 63.2 11.9A a in

Solution

A=

C B

b c

a=

Answers:

.

.

B

b in

c in

29

Example

2. 7.0 9.3a c

Solution

A

C B

b c=

a=

30

Angle of Elevation and Angle of Depression

The angle of elevation for a point above a horizontal line is the angle formed by the horizontal line and the line of sight of the observer at that point.

The angle of depression for a point below a horizontal line is the angle formed by the horizontal line and the line of sight of the observer at that point.

Horizontal line

Horizontal line

Angle of elevation

Angle of depression