Download GCSEAlgebraic Fractions

RIGHT-ANGLED TRIGONOMETRY
Higher Tier
RIGHT ANGLED TRIGONOMETRY

Trigonometry in a right-angled triangle can be used to calculate unknown sides and angles.
Labelling the sides of a Triangle

The hypotenuse (hyp) is always opposite the
right angle.

The opposite (opp) is always opposite the angle
given.

The adjacent (adj) is always next to the angle
given.
The Trigonometric Ratios

The three trigonometric ratios are

These ratios can be remembered by using the word:

These formula triangles are also useful:
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RIGHT-ANGLED TRIGONOMETRY
Higher Tier
Using your calculator

Make sure that your calculator is set to
the degree mode, deg or something
similar should be displayed.
Examples
sin 30° = 05
sin 60° = 08660
cos 45° =07071
tan 70° = 27475

You also need to reverse the process by
using the SHIFT or INV or 2nd F button.
Examples
sin1  0.5  30
cos1  0.7071  45
tan1 2.25  66
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RIGHT-ANGLED TRIGONOMETRY
Higher Tier
Finding the size of an angle
Step 1 Label the sides
Step 2 Decide on the ratio to use
Step 3 Put in the numbers
Step 4 Change the fraction to a decimal
Step 5 Find the angle
EXAMPLE
Find the size of angle 
SOLUTION
Label the sides.
The opposite and the hypotenuse are given, so use sine.
sin 


opp
hyp
8

12
 0.6667

 sin1 0.6667 
 41.8
Important
Don’t round your answers until the very end of the calculation.
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RIGHT-ANGLED TRIGONOMETRY
Higher Tier
Finding the length of a side
EXAMPLE 1
Calculate the length of side AB.
SOLUTION
AB is opposite the given angle. The hypotenuse is
also given. Sine is used in this case.
opp
hyp
AB

12
AB  12 sin 40
 7.71 (2 d.p.)
sin 40 
EXAMPLE 2
Calculate the length of AC.
SOLUTION
The angle and the adjacent are given. AC is the
hypotenuse. Cosine is used in this case.
AC


adj
cos
4.5
cos 25
 4.97
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RIGHT-ANGLED TRIGONOMETRY
Higher Tier
APPLICATIONS

There are many applications of trigonometry – navigation, surveying, building design etc.
Angles of Elevation and Depression
The angle of elevation is
measured from the horizontal
upwards.
The angle of depression is
measured from the horizontal
downwards.
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RIGHT-ANGLED TRIGONOMETRY
Higher Tier
EXAMPLE 1
From a point 55 m away from the base of a building, a man measures the angle of elevation to
the top of the building as 73°. How tall is the building ?
SOLUTION
Draw a sketch.
The angle is given, so is the adjacent.
The opposite is unknown.
h  55  tan73
 180 m
(round answers sensibly)
EXAMPLE 2
From the top of a 28 m cliff, the angle of depression of a boat is 30°. How far is the boat from
the foot of the cliff ?
SOLUTION
28
d
28
d 
 48.5m
tan30
tan30 
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©RSH 13 April 2010
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RIGHT-ANGLED TRIGONOMETRY
Higher Tier
Navigation
EXAMPLE
A ship travels due East from A to B, a distance of 74 km. It then travels due South to C. C is
47 km from B.
a.
What is the bearing of C from A?
b.
What is the distance from A to C
SOLUTION
a.
The required bearing is .
 = 90° + angle BAC.
47
74
 angle BAC = 32° (to the nearest degree)
tanBAC 
 = 90° + 32° = 122°
b.
Using Pythagoras’ theorem
AC² = AB² + BC²
AC² = 74² + 47² = 7685
AC = 88 km (to the nearest km)
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