Download tpc maths (part a) - nswtmth307a

TPC MATHS (PART A) - NSWTMTH307
TOPIC 7: RIGHT ANGLED
TRIGONOMETRY
7.1 Introduction
Trigonometry is just the relationship between angles and sides in triangles. Triangles can
be formed when any two non-parallel lines intersect.
Trigonometry is used to find the missing sides or missing angles in a triangle.
In this topic, we will only deal with right angled triangles, but trigonometry can be used for
any triangle.
Naming the sides of a right angled triangle for trigonometry:
Because trigonometry is the relationship between angles and sides, the naming of the sides
depends on which angle we are using as our focus.
HYPOTENUSE
OPPOSITE
θ
ADJACENT
From the point of view of the angle θ, the sides are named as shown in the figure above.
But if we change the angle that we are interested in, then the opposite and the adjacent
sides swap places (the hypotenuse stays the same) as shown in the figure below.
θ
HYPOTENUSE
ADJACENT
OPPOSITE
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7.2 Decimal Degrees and Degrees, Minutes, Seconds
Angles can be measured in:

Degrees, Minutes, Seconds e.g. 65o37'28" or

Decimal Degrees e.g. 40.5o
Degrees, Minutes, Seconds are smaller and smaller
graduations of a degree. There are:

60 seconds in a minute

60 minutes in a degree
Remember:
There are 90 degrees
in a right angle
Converting between Degrees, Minutes, Seconds and Decimal Degrees can be done using
the o ' " button (sometimes called the bubble button) on the calculator.
When rounding off to the nearest minute, the rule is "if the seconds are 30 or above,
round the minutes up, otherwise leave them the same". Use the same process for
rounding to the nearest degree.
Examples:
Rounding 65o37'28" to the nearest minute becomes 65o37'
Rounding 65o37'28" to the nearest degree becomes 66o
Rounding 14o25'30" to the nearest minute becomes 14o26'
7.3 The Trigonometric Ratios
The trigonometric ratios are Sine, Cosine and Tangent, also known as sin, cos and tan.
They are called trigonometric ratios because they tell us the ratio of two sides of the
triangle. The formulas below describe which sides of the triangle we use for each of the
trigonometric ratios.
sin  
opposite
hypotenuse
cos 
adjacent
hypotenuse
tan  
opposite
adjacent
The mnemonic SOH CAH TOA can be used to help remember the correct equations for the
trigonometric ratios - make sure you know how to spell it, and how to write the equations
from it, otherwise it is useless.
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7.4 Finding a Missing Side in a Right Angled Triangle
To find a missing side in a right angled triangle, follow these steps:
1. Name each of the relevant sides of the triangle (hypotenuse, adjacent or opposite)
from the point of view of the angle
2. Decide which trigonometric ratio to use (think SOH CAH TOA)
3. Write the equation for the trigonometric ratio
4. Fill in the angle and the sides that you know
5. Solve for the missing side
a. if the missing side is on the top of the fraction, multiply the known side by the
sin, cos or tan of the angle
b. if the missing side is on the bottom of the fraction, divide the known side by
the sin, cos or tan of the angle
Example 1: Find the value of the missing side, x in the triangle below.
18m
x
49o
The sides of the triangle that are of interest are the opposite side and the
hypotenuse, so we are dealing with the sin ratio
sin  
opposite
hypotenuse
sin 49o 
x
18
(x is on the top of the fraction, so we multiply to solve for x)
x = 18 x sin 49o
= 13.6m
Example 2: Find the value of the missing side, x in the triangle below.
5.8m
24o
x
The sides of the triangle that are of interest are the opposite side and the
adjacent, so we are dealing with the tan ratio
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tan  
opposite
adjacent
sin 24 o 
5.8
x
(x is on the btm of the fraction, so we divide to solve for x)
x = 5.8 / sin 24o
= 14.3m
7.5 Finding a Missing Angle in a Right Angled Triangle
To find a missing angle in a right angled triangle, follow these steps:
1. Name each of the relevant sides of the triangle (hypotenuse, adjacent or opposite)
from the point of view of the angle that you are trying to find
2. Decide which trigonometric ratio to use (think SOH CAH TOA)
3. Write the equation for the trigonometric ratio
4. Fill in the angle and the sides that you know
5. Solve for the missing angle by using the inverse trig ratio on your calculator (sin-1,
cos-1 or tan-1 which can be found by using SHIFT and then sin, cos or tan)
Example 3: Find the value of the missing angle, θ in the triangle below.
45m
θ
o
26m
The sides of the triangle that are of interest are the adjacent side and the
hypotenuse, so we are dealing with the cos ratio
cos 
adjacent
hypotenuse
cos  o 
26
45
(in finding an angle, we need to use the inverse cos function)
 = cos-1 (26/45)
= 54.7o
= 54o42' (to the nearest minute)
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7.6 Angles of Elevation and Depression
Angles of elevation and depression can be used to calculate heights and distances of
objects in real life using trigonometry.
The important thing to remember is that:

angles of elevation are measured from the horizontal line upwards to the line of
sight

angles of depression are measured from the horizontal line downwards to the line
of sight
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From an observer's position at the top of a 40m high cliff, the angle of depression to an object, P, is
34o. Calculate how far from the base of the cliff the object is.
tan  
opposite
adjacent
tan 34o =
40
x
x = 40 / tan 34o
= 59.3m
The object is 59.3m from the base of the cliff.
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