Angles and Trigonometry
Right Angles
A Right Angle is an angle equal to 90o
Right angles are usually denoted by the little box symbol shown.
Angles in General
| Angles greater than | 0 degrees | and less than | 90 degrees | are called | acute |
 | Angles greater than | 90 degrees | and less than | 180 degrees | are called | obtuse |
 | Angles greater than | 180 degrees | and less than | 360 degrees | are called | reflex |
You can see from the definition of reflex angles that you do need to be aware of which angle
it is that we are interested in, i.e. for a
reflex angle it is the angle greater than 180
degrees as opposed to the obtuse (or acute) angle that would be measured from the 'other side'.
A practical situation where this differentiation is very important is in bearings. Here the angle is measured clockwise from North. So due west would be on a bearing of 270 degrees, definitely not 90 degrees. The bearing shown at left is definitely 220 degrees - not the 140 degrees you would measure if you measured the angle counter-clockwise.
Creating angles of 30, 45, 60, 90
degrees by folding paper
Angles on a Straight Line
The angles on a Straight Line add up to 180o
And the angles in a complete Circle sum to 360o, which you should already have met when learning about Pie Charts.
Angles in a Triangle
The angles in a triangle sum to 180o
Other links
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What is the missing angle in this triangle ? 
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Parallel Lines
Lines which are parallel to each other stay the same distance apart, no matter how long they are.
Opposite Angles
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Vertically opposite angles are equal |
Pythagoras' Theorem
This theorem only works for right-angled triangles
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- Let's Study on Hypotenuses
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The theorem says :
The square of the hypotenuse equals the sum of the squares of the other two sides |
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What is the length of the missing side in these triangles ? |
Sine, Cosine and Tangent
Sines, Cosines and Tangents are defined in terms of the sides of a right-angled triangle. We first give a few definitions regarding the sides of a triangle.
The hypotenuse has already been previously defined - it is
- the longest side in a right-angled triangle
- it is opposite the right-angle
We also require the opposite side and the adjacent side. Whereas the hypotenuse is a inherent property of the triangle, the use of the terms opposite side and adjacent side are relative to which angle we are considering. Consider the following two diagrams
The triangles are identical - hopefully the situation is self-explanatory. The hypotenuse is defined as a property of the triangle itself, but the use of the terms opposite and adjacent will depend on which angle you are considering.
We can now define the sine (sin), cosine (cos) and tangent (tan)
\[ \mbox{cos} = \frac{adjacent}{hypotenuse} \]
\[ \mbox{tan} = \frac{opposite}{adjacent} \]
Hint :- remember the following mnemonic
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Links
Links to Other Sites
Angles and angle terms from Math League
Past Exam Questions
Stage 3
Steve is training for a career in building and is learning how to use a ladder safely.
He has to consider two distances:
- the distance of the foot of the ladder from the wall
- the height of the top of the ladder up the wall.
a) Show that the angle between the ladder and the wall is 14° to the nearest degree.
b) Steve's ladder is 5 metres long.
How far from the wall should he place the foot of the ladder?
Give your answer to an appropriate level of accuracy.
c) Show how you used a different method to check your answer to part b.