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.

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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

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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.

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Angles in a Triangle

The angles in a triangle sum to 180o

Other links

Quick Quiz

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.

When a line crosses a set of parallel lines, there are two sets of angles that we can identify.

Corresponding Angles

Corresponding Angles are Equal

Alternate Angles

Alternate Angles are Equal

Parallelograms are four-sided shapes with opposites sides parallel and equal.

Maths Goodies on Parallelograms


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Opposite Angles

Vertically opposite angles are equal


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Pythagoras' Theorem


This theorem only works for right-angled triangles

First we need to define the hypotenuse. The hypotenuse is the longest side in a right-angled triangle - it is the side opposite the right angle

The theorem says :

The square of the hypotenuse equals the sum of the squares of the other two sides



Stated mathematically
c2 = a2 + b2

Quick Quiz

What is the length of the missing side in these triangles ?
  1.    
  2.    
  3.    



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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)

sin = opposite / hypotenuse

cos = adjacent / hypotenuse

tan = opposite / adjacent

Hint :- remember the following mnemonic

Links

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Angles and angle terms from Math League

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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.

The ratio of these two distances must be 1:4


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.

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