Trigonometric Functions Applied to AC Circuits

Adapted material to accompany Lesson 2315-0

2

IDENTIFYING THE SIDES OF A RIGHT TRIANGLE

• The sides of a right triangle are named the opposite side, adjacent side, and hypotenuse

• The hypotenuse is the longest side of a right triangle and is always opposite the right angle

• The positions of the opposite and adjacent sides depend on the reference angle

• The opposite side is opposite the reference angle

• The adjacent side is next to the reference angle

3

EXAMPLES OF IDENTIFYING SIDES

• The two right triangles below each have their sides labeled according to a given reference angle

• Note: The opposite and adjacent sides vary depending on the reference angle while the hypotenuse stays in the same position in both cases.

A Adjacent

B

Opposite A

dja

cent

4

TRIGONOMETRIC FUNCTIONS

• Three trigonometric functions are defined in the table below

• Note: The symbol denotes the reference angle

FUNCTION SYMBOL DEFINITION

sine θ sin θ

cosine θ cos θ

tangent θ tan θ

hypotenuse

sideopposite

hypotenuse

sideadjacent

sideadjacent

sideopposite

5

RATIO EXAMPLE

1

a

b

c c

a

c

b

b

a

• The sides of the triangle below are labeled with different letters, then each of the listed trigonometric functions are given using 1 as the reference angle

tan 1 =

cos 1 =

sin 1 =

6

DEFINITIONS

• A vector is a quantity that has both magnitude and direction

• Vectors are shown as directed line segments. The length of the segment represents the magnitude and the arrowhead represents the direction of the quantity

• Vectors have an initial point and a terminal point. An arrowhead represents the terminal point

• A vector is named by its two end points or by a single lowercase letter. Arrows are placed above the line segments

7

DEFINITIONS

• Equal vectors have identical magnitudes and directions. A vector can be repositioned provided its magnitude and direction remain the same

• Vectors are usually shown on the rectangular coordinate (x,y) system. A vector is in standard position when the initial point is at the origin of the rectangular coordinate (x,y) system and its angle is measured counterclockwise from the positive x-axis

8

• The vector shown below could be named either

• Vectors below are not equal vectors because they do not have identical magnitudes and directions.

EXAMPLES

.or fAB

A

B

f

ba and

a b

9

VECTOR NOTATION

• A vector can be represented on the rectangular coordinate system using ordered pair notation (x,y), or vectors can be shown and solved as lengths and angles. Note: Vector

angles are often represented as (theta)

• The vector shown on the following slide could either be read as 3x + 4y [(3,4)cm] in the rectangular coordinate system or as 5cm at

53.1° ( 32 + 42 = 5)

10

VECTOR NOTATION

Circuit for example problem

Problem, part 1:

Answer:

Given that this is a series circuit, all answers should be 2A regardless of the component used. The one thing we should be able to count on is the current is the same through all components.

• In the previous figure, how much current is flowing

a) Through the 30Ω resistor?

b) Through the 40Ω capacitive reactance, XC?

c) To and from the terminals of the applied voltage, VT?

Problem, part 2:

• In the previous figure, what is the phase relationship between

a) I and VR?

b) I and VC?

c) VR and VC?

Answers:

a) Should be zero degrees as resistor is passive component

b) VC should lag by 90 degrees

c) VC should lag by 90 degrees

Problem, part 3:

• In the previous figure, how much is the applied voltage, VT?

• How much is the total impedance, ZT?

• Answers:

VT = 602 + 802 =

3600 + 6400 =

10000 = 100 volts

ZT = 302 + 402 =

900 + 1600 = 2500 = 50 ohms

Problem, part 4:

• Draw the phasor voltage triangle for the previous figure (use VR as the reference phasor; show two ways).

Vector representations

VR VR

VC VT VC

VT