Geometry and trigonometry
This page addresses key concepts within trigonometry, which is the study of triangles and their angles.
In brief, it covers:
- Different types of triangles
- Pythagoras’ theorem
- What similar triangles are and how to identify them
- The sine, cosine and tangent trigonometric ratios
- The sine rule
- The cosine rule
- Which formula to use when working with triangles
Triangles
A triangle is a shape with three sides. In mathematics, the angles of a triangle are usually labelled with capital letters, while the sides are labelled with lower case letters. Side a will always be opposite angle A, side b will be opposite angle B, and side c will be opposite angle C.
If the sides are not labelled with lower case letters, they can be named using the points at each end, such as AB, BC, and AC. The angles can also be written as \(\angle A\), \(\angle B\) and \(\angle C\).
An angle may also be named using three letters that show the points of the triangle. For example:
- \(\angle A\) can also be written as \(\angle BAC\)
- \(\angle B\) can also be written as \(\angle ABC\)
- \(\angle C\) can also be written as \(\angle ACB\).
When an angle is named with three letters, the angle is always at the middle letter.
The angles inside any triangle always add up to \(180^\circ\).
There are six types of triangles:
Acute triangles contain angles that are each less than \(90^\circ\).
Obtuse triangles contain one angle that is more than \(90^\circ\). The other two angles will be acute angles. You can see in the below triangles that angles B and D are both more than \(90^\circ\), while A, C, E and F are all acute angles.
Right-angled triangles contain an angle that is equal to \(90^\circ\). This angle is usually marked by a little square in that corner.
All angles within an equilateral triangle are the same value, and all sides are the same length. As all angles within a triangle add up to \(180^\circ\), each angle in an equilateral triangle will be \(60^\circ\).
Isosceles triangles have two equal sides and two equal angles. The equal sides below are shown using lines on each matching side.
In scalene triangles all the angles are different sizes and all of the sides are different lengths. Acute, obtuse, and right-angled triangles can all be scalene triangles as well, as seen below. The triangle on the left is both a scalene triangle and an acute triangle, as all the angles are less than \(90^\circ\), while the triangle on the right is both scalene and obtuse as angle E is larger than \(90^\circ\).
Check your understanding of different types of triangles by working through the activity:
Pythagoras’ theorem
Pythagoras’ theorem shows the relationship between the lengths of the sides in a right-angled triangle. It can be used to find the length of a missing side when the lengths of the other two sides are known.
Consider the right-angled triangle shown below:
Pythagoras’ theorem tells us that:
\[a^2 + b^2 = c^2\]In the formula, the side c is always the hypotenuse. The hypotenuse is the longest side of a right-angled triangle and is always opposite the right angle.
If we know the values of two of the sides, we can use Pythagoras’ theorem to find the value of the third side.
Example 1: Given the triangle below, find the value of \(\chi\).
Solution:
\[\begin{aligned} a^2 + b^2 &= c^2 \\\ 9^2 + 12^2 &= \chi^2 \\\ 81 + 144 &= \chi^2 \\\ 225 &= \chi^2 \\\ \sqrt{225} &= \chi \\\ \chi &= 15cm \\\ \end{aligned}\]Example 2: Given the triangle below, find the value of y to 2 decimal places.
Solution:
\[\begin{aligned} a^2 + b^2 &= c^2 \\\ a^2 &= c^2 - b^2 \\\ y^2 &= 25^2 - 17^2 \\\ y^2 &= 336 \\\ y &= \sqrt{336} \\\ y &= 18.33cm \\\ \end{aligned}\]Note: We can rearrange our equations using algebra.
Check your understanding of Pythagoras’ theorem using the questions below.
Similar triangles
Two triangles are similar if they have the same shape. This means their corresponding angles are equal. However, similar triangles do not have to be the same size. If two similar triangles are also the same size, they are called congruent triangles.
To show that two triangles are similar, you must show that one of the following conditions is true:
- Two pairs of corresponding angles are equal
- Three pairs of corresponding sides are proportional
- Two pairs of corresponding sides are proportional and the corresponding angles between them are equal
If one of these conditions is met, the triangles are similar.
You can think of similar triangles as enlarged or reduced versions of the same shape.
Example 1:
The two triangles below are similar as their corresponding sides are proportional (the sides of the smaller triangle are half the size of the larger triangle). You could rotate the triangle on the right to make it look the same as one on the left.
Example 2:
The two triangles below are similar triangles as their corresponding angles are equal in size. Both the large triangle and the smaller, purple, triangle share the \(\angle B\). In addition, as lines \(DE\) and \(AC\) are perpendicular, \(\angle A\) and \(\angle D\) are equal, and \(\angle C\) and \(\angle E\) are equal.
Example 3:
Given the two triangles below are similar, find the value of lines \(JL\) and \(KL\).
Solution:
To find line \(JL\):
\[\begin{aligned} \frac{GH}{JK} &= \frac{GI}{JL} \\\ \frac{8}{12} &= \frac{10}{JL} \\\ 8 \times JL &= 120 \\\ JL &= 15 \\\ \end{aligned}\]To find \(KL\):
\[\begin{aligned} \frac{GH}{JK} &= \frac{HI}{KL} \\\ \frac{8}{12} &= \frac{9}{KL} \\\ 8 \times KL &= 108 \\\ JL &= 13.5 \\\ \end{aligned}\]Note: We can rearrange our equations using algebra.
You can check your understanding of similar triangles by working through the questions below.
Trigonometric ratios
There are six trigonometric functions that describe the relationship between the acute angles in a right-angled triangle and the length of its sides. However, in most cases, you will only need to use three of them:
- Sine (\(\sin\))
- Cosine (\(\cos\))
- Tangent (\(\tan\))
While using trigonometric functions, the angle you are working with is usually represented by the symbol \(\theta\), known as theta.
Each side of a right-angled triangle has a specific name. The longest side, which is always opposite the right angle, is called the hypotenuse. The other two sides are called the opposite side and the adjacent side.
The opposite side is the side directly opposite the angle being used in the calculation. The adjacent side is the side next to the angle being used, excluding the hypotenuse.
We can use trigonometric functions to calculate the relationship between an angle in a right-angled triangle and the lengths of its sides. These relationships are called trigonometric ratios.
To find the sine of an angle, divide the length of the opposite side by the length of the hypotenuse.
To find the cosine of an angle, divide the length of the adjacent side by the length of the hypotenuse.
To find the tangent of an angle, divide the length of the opposite side by the length of the adjacent side.
These rules are written:
\[\sin\theta = \frac{opp}{hyp}\] \[\cos\theta = \frac{adj}{hyp}\] \[\tan\theta = \frac{opp}{adj}\]You can use the acronym SOHCAHTOA to help you remember which sides are used with each function:
Example: Given the triangle below, find the values of angles B and C to two decimal places.
Solution: For \(\angle B\):
First, work out which side is the opposite and which is the adjacent. The longest side, opposite the right angle, is always the hypotenuse.
Side AC is opposite angle B. Side AB is adjacent. The hypotenuse is BC. We don’t know the value of AB, so we’re going to use the sine function.
\[\sin\theta = \frac{opp}{hyp}\] \[\sin B = \frac{12}{15}\] \[\sin B = 0.8\]We can then use the inverse sine function on a calculator (sometimes called arc sine) to find the angle whose sine is equal to \(0.8\):
\[\begin{aligned} B &=\sin^{-1} 0.8 \\\ B &= 53.13^\circ \\\ \end{aligned}\]We can now find \(\angle C\) in two different ways:
One way is by using a trigonometric function. This time, AC is the adjacent side. As we know the values of the adjacent and hypotenuse, we’re going to use the cos function.
\[\begin{aligned} \cos\theta &= \frac{adj}{hyp} \\\ \cos C &= 0.8 \\\ C &= \cos^{-1} 0.8 \\\ C &= 36.87^\circ \\\ \end{aligned}\]Another way to find \(\angle C\) is by using the fact that all angles in any triangle add up to \(180^\circ\).
\[180 - 90 - 53.13 = 36.87^\circ\]Check your understanding of how to use trigonometric ratios by working through the questions below.
Sine rule
The sine rule can be used to solve problems involving any triangle when at least one side and its opposite angle are known. It can be used to find a missing side length or a missing angle.
When we’re given a triangle, for example:
We can find the value of a side of a triangle using the sine rule:
\[\frac{a}{sin\;A} = \frac{b}{sin\;B} = \frac{c}{sin\;C}\]To find the value of an angle, rearrange the sine rule so that the angles are on the top:
\[\frac{sin\;A}{a} = \frac{sin\;B}{b} = \frac{sin\;C}{c}\]To use the sine rule you will need to know at least one pair of a side with its opposite angle, and you will only need to use two parts of the sine rule, not all three.
Example 1: Find the value of K to two decimal places.
Solution:
\[\begin{aligned} \frac{a}{sin\;A} &= \frac{b}{sin\;B} \\\ \frac{K}{sin\;50^\circ} &= \frac{21}{sin\;85^\circ} \\\ K &= \frac{21\,sin\;50^\circ}{sin\;85^\circ} \\\ K &= 16.15cm \\\ \end{aligned}\]Example 2: Find the value of \(\angle \theta\) to two decimal places.
Solution:
\[\begin{aligned} \frac{sin\;A}{a} &= \frac{sin\;B}{b} \\\ \frac{sin\;\theta}{57} &= \frac{sin\;35^\circ}{42} \\\ sin\;\theta &= \frac{57\,sin\;35^\circ}{42} \\\ sin\;\theta &= 0.778425 \\\ sin^{-1} 0.778425 &= 51.116 \\\ \theta &= 51.12^\circ \\\ \end{aligned}\]Check your understanding of the sine rule by working through the questions below.
Cosine rule
The cosine rule can be used with any triangle when you need to find a side length or angle using information about all three sides and one angle. It describes the relationship between the lengths of the sides and an angle in a triangle.
Consider the triangle below:
The cosine rule tells us:
\(a^2 = b^2 + c^2 - 2bc\) \(cos\;A\)
To find the value of the angle rather than the value of a side, we can rearrange the formula:
\[cos\;A = \frac{b^2 + c^2 - a^2}{2bc}\]Remember that the \(\angle A\) will always be opposite side a.
Example 1: Find the value of a, to one decimal place.
Solution:
\[\begin{aligned} a^2 &= b^2 + c^2 - 2bc\;cos\;A \\\ a^2 &= 8^2 + 14^2 - 2 \times 8 \times 14cos\;120^\circ \\\ a^2 &= 64 + 196 - 224 \times (-0.5) \\\ a^2 &= 372 \\\ a &= \sqrt 372 \\\ a &= 19.3 m \\\ \end{aligned}\]Example 2: Find the value of \(\angle \theta\) to two decimal places.
Solution:
\[\begin{aligned} cos\;A &= \frac{b^2 + c^2 - a^2}{2bc} \\\ cos\;\theta &= \frac{12^2 + 14^2 - 13^2}{2 \times 12 \times 14} \\\ cos\;\theta &= \frac{12^2 + 14^2 - 13^2}{2 \times 12 \times 14} \\\ cos\;\theta &= \frac{171}{336} \\\ cos\;\theta &= 0.50893 \\\ \theta &= cos^{-1}\,0.50893 \\\ \theta &= 59.41^\circ \\\ \end{aligned}\]Check your understanding by working through the questions below.
What formula should I use?
