Download Sine

Contents
Introduction
3
Trigonometry
4
Sides and angles of a right triangle
4
The Trigonometric ratios
5
Sine
5
Plotting the sine function
6
Cosine
7
Tangent
8
Computing sides and angles
8
Pythagoras’ theorem
12
Summary
14
Answers
18
Check your progress
EGG202A: Appendix 1 Apply trigonometry and Pythagoras' Rule
 NSW DET 2017 2006/060/04/2017 LRR 3818
19
1
2
EGG202A: Appendix 1 Apply trigonometry and Pythagoras' Rule
 NSW DET 2017 2006/060/04/2017 LRR 3818
Introduction
This section is an introduction to the basic maths used in ac circuit
problems. This may be revision for some students but it is important to have
a thorough understanding of the subject matter.
A great deal of the later work relies on this, as explanations of circuit
conditions and calculations in ac circuits use basic trigonometry.
In the following section, we will solve problems involving right-angled
triangles and use the right-angled triangle to explain the property that
opposes current in ac circuits.
At the end of this section you should be able to:

state and apply the sine, cosine and tangent ratios of a right angle
triangle

determine the correct trigonometric formulae to use in ac circuit
calculations
EGG202A: Appendix 1 Apply trigonometry and Pythagoras' Rule
 NSW DET 2017 2006/060/04/2017 LRR 3818
3
Trigonometry
Trigonometry is the mathematics which deals with the angles and sides of
triangles. In this unit we will only look at the right-angled triangle. A rightangled triangle has one angle of 90(a right angle).
Sides and angles of a right triangle
In right-angled triangles we name the sides in relation to the acute angle we
are considering. These sides are labelled hypotenuse, adjacent and opposite,
as in Figure 1.
Figure 1: Right-angled triangle
The hypotenuse is the side opposite the right angle, the adjacent side is the
side alongside (adjacent to) the acute (less than 90°) angle we are interested
in, while the side opposite this angle is called the opposite side.
In ac circuit theory, these sides are also represented by symbols, Z for the
hypotenuse, R for the adjacent side and X for the opposite side. These
symbols will be explained when we see the use of the triangle in solving
problems in ac circuits.
4
EGG202A: Appendix 1 Apply trigonometry and Pythagoras' Rule
 NSW DET 2017 2006/060/04/2017 LRR 3818
The Trigonometric ratios
There are three main trigonometric ratios, based upon the sides of the right
angle triangle. They are the sine, cosine, and tangent ratios.
A ratio is the relationship of one number to another - that is, a proportion or
fraction.
The three trigonometry ratios we need from the right-angled triangle are
found from the ratio of two sides of the triangle in relation to the acute angle
we are considering. These ratios are now introduced.
Sine
The sine of an angle is the ratio of the opposite side to the hypotenuse. So in
Figure 1, the sine of angle  (phi) is:
opposite
hypotenuse
X
sin  
Z
sin  
The sine ratio for any angle may be found using a scientific calculator. First
enter the angle, say 30°, then press the key labelled sin. The result is 0.5.
Notes:

The exact key sequence will depend upon your brand of calculator.
For example, you may need to press a ‘secondary function’ key
before you press the sin key. You must become familiar with the
operation of your calculator by consulting its user guide.

We will always work in degrees, but calculators can use a number of
modes for the trig functions, including degrees and radians. Be sure
that the mode of your calculator is set to degrees.
Values of the sin function are between minus one and positive one. For
angles between zero and 180° the sine ratios are positive while between
180° and 360° the ratios are negative.
EGG202A: Appendix 1 Apply trigonometry and Pythagoras' Rule
 NSW DET 2017 2006/060/04/2017 LRR 3818
5
Student exercise 1
Using a calculator, find the sine of the following angles:
1
0°, 15°. 30°, 60°, 100°
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
2
190°, 240°, 270°, 330°
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
Check your answers with those given at the end of the section.
Plotting the sine function
Using this information on the sine of a range of angles, we can plot the sine
function from 0 to 360° in 15 degree steps, the angles plotted on the
horizontal axis and the sine values on the vertical axis. The result is shown
in Figure 2.
Note that this graph shows only one cycle of the sine function. The curve
actually continues forever for all negative and positive angles.
6
EGG202A: Appendix 1 Apply trigonometry and Pythagoras' Rule
 NSW DET 2017 2006/060/04/2017 LRR 3818
Figure 2: Plotting the sine wave
Cosine
The cosine of an angle-is the ratio of the adjacent side to the hypotenuse.
So in Figure 1:
adjacent
hypotenuse
R
cos  
Z
cos  
To find a cosine ratio on a calculator, enter the angle, say 45°, and press the
key labelled ‘cos’. The result is 0.707.
Cosine values are between one and zero or between zero and minus one.
Cosine values for angles between zero and 90° are positive, while angles
between 90° and 270° are negative. For angles between 270° and 360° the
values again are positive.
If we plot a curve of cosine values from 0° to 360°, the result will appear as
that shown in Figure 3.
EGG202A: Appendix 1 Apply trigonometry and Pythagoras' Rule
 NSW DET 2017 2006/060/04/2017 LRR 3818
7
Figure 3: Cosine wave
The curve in Figure 3 is sinusoidal in form, but displaced 90° from the
position we found the sine wave to be.
Tangent
The tangent of an angle is the ratio of the opposite side to the adjacent side.
So in Figure 1:
opposite
adjacent
X
tan  
R
tan  
To find a tangent ratio on a calculator, enter the angle, say 20°, and press the
key labelled tan. The result is 0.36397.
Tangent values are between zero and infinity. Tan 0° is zero and tan 90° is
infinity. Your calculator will give an error for tan 90°, as the number is too
large to be computed.
Computing sides and angles
Example:
From the data given in Figure 4, determine the length of the sides labelled X
and R.
8
EGG202A: Appendix 1 Apply trigonometry and Pythagoras' Rule
 NSW DET 2017 2006/060/04/2017 LRR 3818
Figure 4: Determining the lengths of sides
Solving for X
The data given is the length of the hypotenuse Z and the acute angle . As X
is the opposite side to the angle and the sine ratio uses the opposite side and
the hypotenuse, the equation for sin  can be used.
opposite
hypotenuse
X

Z
sin  
Make X the subject by multiplying both sides by Z.
sin   Z  X
 X  Z  sin 
X  Z  sin 
 200  0.5
 100 mm
Solving for R
As R is the adjacent side to  and the length of the hypotenuse is known, the
cos ratio may be used.
adjacent
hypotenuse
R

Z
cos  
Make R the subject by multiplying both sides by Z:
cos   Z  R
Therefore:
EGG202A: Appendix 1 Apply trigonometry and Pythagoras' Rule
 NSW DET 2017 2006/060/04/2017 LRR 3818
9
R  Z  cos 
 200  0.866
 43.3 mm
The trigonometric ratios of the angle in the right-angled triangle can also
determine the angle, if any ratio is known.
If the sine ratio of an angle is known, the size of the angle may be found
using the inverse sine or arcsine function on a calculator. The inverse sine
function gives the angle whose sin is the number specified.
The inverse sine function is written sin – 1. To use this on a calculator, first
press the inverse key, then the sin key. (Again, the details of your calculator
may vary – check your handbook for details.)
Example:
The sine of an angle is given as 0.75 and we wish to know the size of the
angle in degrees. First enter the sine into the calculator, and then follow with
the keystrokes inv sin. The result is 48.59°.
Likewise, the arcos function is used to find the angle if the cosine ratio is
known. Arcos means ‘the angle whose cosine is...’
Example:
The cosine of an angle is given as 0.866. To find the angle, enter the cosine
in a calculator then use the keystrokes inv cos. The result is 30°.
There is a similar function available for tangent called arctan.
10
EGG202A: Appendix 1 Apply trigonometry and Pythagoras' Rule
 NSW DET 2017 2006/060/04/2017 LRR 3818
Student exercise 2
1
Find the angles whose sine ratio is equal to:
(a) 0.5
___________________________________________________________________
(b) 0.9
___________________________________________________________________
(c) 1.0
___________________________________________________________________
2 Find the angles whose cosine ratio is:
(a) 0.3
___________________________________________________________________
(b) 0.75
___________________________________________________________________
(c) 1.0
___________________________________________________________________
Check your answers with those given at the end of the section.
EGG202A: Appendix 1 Apply trigonometry and Pythagoras' Rule
 NSW DET 2017 2006/060/04/2017 LRR 3818
11
Pythagoras’ theorem
Pythagoras’ theorem relates the lengths of the sides of a right-angled
triangle to each other. The theorem states:
‘In a right-angled triangle the square on the hypotenuse is equal to the
sum of the squares on the other two sides.’
This is written mathematically as:
Z2 = R2 + X2
We use the terms Z, R and X because they represent ac circuit quantities:

Z = impedance which is total opposition to current

R = resistance (the opposition to current from resistors)

X = reactance (the opposition to current from ideal capacitors and
inductors)
A graphical representation is shown in Figure 5.
12
EGG202A: Appendix 1 Apply trigonometry and Pythagoras' Rule
 NSW DET 2017 2006/060/04/2017 LRR 3818
Figure 5: Graphical representation of Pythagoras’s theorem
From the initial equation, the following three equations may be derived:
Z  R2  X 2
R  Z2  X 2
X  Z 2  R2
Student exercise 3
Determine the length of the unknown side in the triangles in Figure 6.
Figure 6
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
Check your answers with those given at the end of the section.
EGG202A: Appendix 1 Apply trigonometry and Pythagoras' Rule
 NSW DET 2017 2006/060/04/2017 LRR 3818
13
Summary

The sides of right-angled triangles are named hypotenuse, adjacent
and opposite to a given enclosed angle. Symbols used in this unit to
represent the sides are:
o Z: the hypotenuse
o R: the adjacent
o X: the opposite.

The three trigonometric ratios used in the unit are sine, cosine, and
tangent. They are defined by the ratios:
X
Z
R
cos  
Z
X
tan  
R
sin  

The inverse trigonometric ratios give the angle from the number that
is the sine, cosine or tangent of that angle:
o Arcsine means the angle whose sine is…
o Arcos means the angle whose cosine is…

Pythagoras’s theorem may be expressed as:
Z 2  R2  X 2
where:
Z = impedance
R = resistance
X = reactance
Therefore:
Z  R2  X 2
14
EGG202A: Appendix 1 Apply trigonometry and Pythagoras' Rule
 NSW DET 2017 2006/060/04/2017 LRR 3818
Check your progress
Part A
In questions 1–12, place the letter matching your answer in the brackets provided.
1
In a right angle triangle the ratio of opposite side to adjacent side is equal to:
(a) unity
(b) sine
(c) cosine
(d) tangent
2
(
)
(
)
(
)
(
)
(
)
The sine of an angle is positive between:
(a) 0° and 90°
(b) 0° and 180°
(c) 180° and 360°
(d) 0° and 90° and between 270° and 360°
3
In a right-angled triangle the trigonometrical ratio of the opposite side to the
hypotenuse is the:
(a) sine
(b) cosine
(c) tangent
(d) cotangent
4
The hypotenuse of a right-angled triangle, of sides 80 and 60 millimetres is:
(a) 20 mm
(b) 100 mm
(c) 140 mm
(d) 480 mm.
5
The length of one side of a right-angled triangle which has a hypotenuse of
130 millimetres and the other side of 50 millimetres is:
(a) 20 mm
(b) 120 mm
(c) 180 mm
(d) 650 millimetres
EGG202A: Appendix 1 Apply trigonometry and Pythagoras' Rule
 NSW DET 2017 2006/060/04/2017 LRR 3818
15
6
The cosine of 150° is equal to:
(a) – 0.866
(b) + 0.5
(c) + 0.866
(d) 1
7
(
)
(
)
(
)
(
)
(
)
(
)
The sine of an angle is the ratio:
(a) opposite/adjacent
(b) adjacent/hypotenuse
(c) hypotenuse/adjacent
(d) opposite/hypotenuse
8
The complement of an angle of 45° is an angle of:
(a) 25°
(b) 30°
(c) 45°
(d) 135°
9
The values of the tangent trigonometrical ratio ranges between:
(a) 0 and 1
(b) 0 and 90
(c) 0 and 180
(d) 0 and  .
10 Arcos means:
(a) the supplement of cosine
(b) the angle whose cosine is
(c) a ratio consisting of sines
(d) a rational coordinate of squares.
11 In a circuit containing resistance only:
(a) current lags voltage by 90°
(b) current leads voltage by 90°
(c) voltage leads current by 90°
(d) the current and the voltage are in phase.
16
EGG202A: Appendix 1 Apply trigonometry and Pythagoras' Rule
 NSW DET 2017 2006/060/04/2017 LRR 3818
Part B
1
Determine the sine, cosine and tangent ratios of the following angles:
20°, 30°, 45°, 90°, 195°, 270°
2
On a sheet of graph paper, using the same axes, plot the curves for a:
(a) sine wave having a maximum value of 60 units
(b) cosine wave having a maximum value of 60 units, for angles from 0° to 360°.
Use a scale of 1 millimetre equals 1 unit and one millimetre equals 3°.
3
Determine the third side of a right-angle triangle, given the hypotenuse as 13 units
and the other side 12 units.
4
A right-angled triangle has sides of the following values: 6, 8 and 10.
Without using a calculator, and showing all working, determine the sine, cosine and
tangent for the two acute angles.
5
Determine the two acute angles in the triangle of Question 4
Figure 8: Diagram for question 6
6
Determine the length of the line AB in the diagram in Figure 8.
Data: φ1 = 60°
2 = 30°
Hint: To find AC, use the tangent ratio for the small triangle.
Answers to Check your progress are at the end of the module.
EGG202A: Appendix 1 Apply trigonometry and Pythagoras' Rule
 NSW DET 2017 2006/060/04/2017 LRR 3818
17
Answers
Student exercise 1
1
Zero, 0.2588, 0.5, 0.866, 0.9848.
All these are positive numbers between 0 and 1.
2
0.1736, – 0.866, –1, – 0.5.
All these are negative numbers between 4 and –1.
Student exercise 2
1
30°, 64.16°, 90°
2
72.54°, 41.4°, 0°
Student exercise 3
1
Z  R2  X 2
 102  7 2
 12.21
2
R  Z2  X 2
 252  162
 19.21
3
X  Z 2  R2
 562  102
 55.1
18
EGG202A: Appendix 1 Apply trigonometry and Pythagoras' Rule
 NSW DET 2017 2006/060/04/2017 LRR 3818
Check your progress
Part A
1
(d)
6
(a)
2
(b)
7
(d)
3
(a)
8
(c)
4
(b)
9
(d)
5
(b)
10 (b)
Part B
1
Angle
Sine
Cosine
Tangent
20
0.342
0.94
0.364
30
0.5
0.866
0.577
45
0.707
0.707
1
90
1
0
infinity
195
– 0.259
– 0.966
0.268
270
–1
0
infinity
2
as specified
3
17.7
4
sin1 = 6/10, cos1 = 8/10, tan 1 = 6/8
sin2 = 8/10, cos2 = 6/10, tan2 = 8/6
5
30, 60
6
82.3
EGG202A: Appendix 1 Apply trigonometry and Pythagoras' Rule
 NSW DET 2017 2006/060/04/2017 LRR 3818
19