Chapter 4

The Trigonometric
Functions

4-1

Angles

„ Defining an angle:
Ray

T
Vertex
Initial side

4-2
 Angle Conversions

„ 360° = 2S radians = 1 revolution

„ 1 degree = 60 minutes
„ 1 minute = 60 seconds There are 3600
sec in 1 deg.

4-3

Angle Conversions

„ Example 1:
„ Convert 65q25’ to decimal degree form.

25
65 q25' 65 q  65 .42 q
60 Divide
by 60.

4-4
 Angle Conversions

„ Example #2
„ Convert 32.459q to degrees, minutes,
seconds.
32.459q
Multiply
32q  0.459 u 60 by 60.
32q  27.54'

4-5

Angle Conversions

„ Example #2 (continued)
„ Convert 32.459q to DMS form.

32.459q
32q27'0.54 u 60
32q27'32.4"
4-6
Angle Conversions

„ Formulas to Convert:
„ Use the fact that there are 360° in 2S
rads in a full circle.
„ Multiply angle in S
degrees to find rads by: 180
„ Multiply angle in 180
rads to find degrees by: S
4-7

Standard Position of an Angle

„ If the initial side of the angle is the positive x-
axis, and the vertex is the origin, the angle is
said to be in standard position.

y-axis

T
x-axis
4-8
Standard Position of an Angle
„ The terminal side of an angle is uniquely
determined by knowing that it passes
through the point (x, y).

(x, y)
y-axis

T
x-axis W
4-9

Determining the
Trigonometric Ratios

We label the
right triangle
as: y
r
y

‘A x x

The trigonometric ratios are defined as

follows:
4-10
 The Trigonometric Functions
r
„ Sine of T: sin T
y „ Cosecant of T: csc T
r y

x r
„ Cosine of T: cos T „ Secant of T: sec T
r x

y x
„ Tangent of T: tan T „ Cotangent of T: cot T
x y

4-11

Evaluating the Trigonometric

Functions

„ The values of a trigonometric function

are dependent on:
{ The ratios of the sides
{ The Pythagorean Theorem

r2 x2  y2

4-12
 Ch. 4.3: Values of the
Trigonometric Functions

„ If given a point on the plane, we can

determine the trigonometric ratios of the
angle made by the terminal side defined
by that same point.
„ There are some angles whose
trigonometric values you should be
familiar with.

4-13

Values of the Trigonometric

Functions

60˚
„ The 30˚-60˚-90˚ triangle: 2 1

30˚ 90˚
—3

45˚
„ The 45˚-45˚-90˚ triangle: —2 1

45˚ 90˚
W 1

4-14
 Finding Unknown Angles
„ We determine the unknown angles
using the inverse trigonometric keys of
the calculator.
„ The notation used for the ratios is:

1 y 1 x 1 y
T sin T cos T tan
r r x
„ Another commonly used notation is
arcsin, arccos and arctan.
4-15

Finding Unknown Angles

„ If sin A = 0.496 then A = sin-1 0.496
„ Before you begin, do you want the
angle in degrees? If so, make sure your
calculator is on degree mode!
„ A = sin-1 0.496 = 29.74°

4-16
Ch. 4.4: The Right Triangle

„ We can generalize the definitions of the

trigonometric functions by naming the
sides of a right angle triangle.
B

Side opposite A

A
C
Side adjacent A
4-17

Procedure for Solving a Right

Triangle

1. Sketch a right triangle and label the

known and unknown sides and angles.
2. Express each of the 3 unknown parts in
terms of the known parts and solve for
the unknown parts.
3. Check the results.

4-18
Procedure for Solving a Right
Triangle

„ The sum of the angles should be 180˚.

„ If only one side is given, check the
computed side with the Pythagorean
Theorem.
„ If 2 sides are given, check the angles
and computed side by using
appropriate trigonometric functions.
W
4-19

Ch. 4.5: Applications of Right

Triangles

„ Here we introduce the ideas of:

{ the angle of depression, and,
{ the angle of elevation.

4-20
Angle of Elevation

„ The angle between the line of sight (line

joining the eye of an object) & the
horizontal plane with the object ABOVE
the horizontal plane.

A Worm’s View
Looking Up!
T
4-21

Angle of Depression

„ The angle between the line of sight (line

joining the eye of an object) & the
horizontal plane with the object BELOW
the horizontal plane.

T A Bird’s View
Looking Down!

4-22
 Working with Angle of Depression
„ In the diagram below, an observer on top of
building A (293 m high) measures the angle of
depression to the bottom of B as 62.6°. How far
away is building B?

293 m
Distance?
A B
4-23

Working with Angle of Depression

„ We sketch out a diagram labelling all the

pertinent information. Notice where we
measure 62.6°.
62.6°

293 m
62.6°
4-24
Working with Angle of Depression

„ If we remove the buildings, a right angle

triangle has been created.
„ We solve for
distance having
identified the
appropriate
293 m

trigonometric ratio.
Distance? 62.6°
4-25

Working with Angle of Depression

293
tan 62 .6 q
d
293
d 151 .87 m
tan 62 .6 q

62.6°
Distance 4-26
Working with Angle of Depression

„ Therefore, the buildings are 151.87 m apart.

151.87 m
W
4-27

Example 1

„ Find the gear angle θ, if t = 0.18 cm

4-28
Example 2

„ A coast guard boat 2.75 km from a straight

beach can travel at 37.5 km/h. By travelling
along a line that is 69° with the beach, how
long will it take the boat to reach the
beach?

4-29

Example 3
„ Part of the Tower Bridge in London is a drawbridge.
This part of the bridge is 76 m long. When each half
is raised, the distance between them is 8 m. What
angle does each half make with the horizontal?

4-30
Example 4
„ A square wire loop is rotating in the magnetic field between
two poles of a magnet in order to induce an electric current.
The axis of rotation passes through the center of the loop and
is midway between the poles, as shown in figure below. How
far is the edge of the loop from either pole if the side of the
square is 7.3 cm and the poles are 7.66 cm apart when the
angle between the loop and the vertical is 78.0°?

4-31

Example 5
„ A table top is in the shape of a regular octagon (eight sides).
What is the largest distance across the table, if one side of the
octagon table is 0.75 m?

4-32