Trigonometry is the study of triangles and their angles. This page covers key concepts within trigonometry including Pythagoras’ theorem, trigonometric ratios and the sine and cosine rules.

Triangles

A triangle is a figure that has three sides. Generally, the angles of a triangle are represented using capital letters, and the sides are represented by 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.

A triangle with angles marked A, B, and C. It's sides are marked lower case a, b, and c.

If the sides aren’t labelled, we can refer to them as AB, BC, and AC. The angles may also be referred to as \(\angle A\), \(\angle B\) and \(\angle C\). Each angle may also be referred to in relation to the labelled points of the triangle. For example, \(\angle A\) could also be \(\angle BAC\), \(\angle B\) could also be \(\angle ABC\), and \(\angle C\) can be written as \(\angle ACB\). If this is the case, notice the angle you are working with will always be at the letter in the middle.

The sum of the angles inside any triangle will always be \(180^\circ\).

There are six types of triangles:

Acute triangles contain angles that are each less than \(90^\circ\).

Two triangles with angles that are all less than ninety degrees.

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.

Two triangles, each containing one angle that is over ninety degrees.

Right-angled triangles contain an angle that is equal to \(90^\circ\). This angle is usually marked by a small box in that corner.

Two triangles, each containing one angle that is ninety degrees. These ninety degree angles are marked with a box in the 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 \(60^\circ\).

Two triangles with angles that are all less than ninety degrees.

Isosceles triangles have two equal sides and two equal angles. The equal sides below are shown using lines on each matching side.

A triangle  with two equal sides, marked by lines, and two equal angles.

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 \(90^\circ\).

Two scalene triangles, one with acute angles, one with two acute angles and an obtuse angle.

Check your understanding of different types of triangles by working through the activity:

Pythagoras’ theorem

Pythagoras’ theorem shows us that the values of all the sides in a right-angled triangle are related. Given the triangle below:

A right-angled triangle. The sides are labelled a, b, and c, c being the hypotenuse.

Pythagoras’ theorem tells us that:

\[a^2 + b^2 = c^2\]

In the formula, the side c is always the hypotenuse.

If we know the values of two of the sides, we can use Pythagoras’ theorem to find the unknown third side.

Example 1: Given the triangle below, find the value of \(\chi\).

A right-angled triangle. The sides are labelled 9cm, 12cm, and x, x being the hypotenuse.

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.

A right-angled triangle. The sides are labelled 17cm, y, and 25cm, with 25cm being the hypotenuse.

Solution:

\[\begin{aligned} a^2 + b^2 &= c^2 \\\ a^2 &= c^2 - b^2 \\\ y^2 &= 25^2 - 17^2 \\\ y^2 &= 914 \\\ y &= \sqrt{914} \\\ y &= 30.23cm \\\ \end{aligned}\]

Note: We can rearrange our equations using algebra.

Check your understanding of Pythagoras’ theorem using the questions below.

Similar triangles

Two triangles are considered similar if they have the same shape, in terms of having the same angles. However, they don’t necessarily need to be the same size (if they are actually the same size, this is a special kind of similar triangle called a congruent triangle). To establish that two triangles are similar you need to show that one of the following criteria are met (as then the rest will follow):

  • 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

You could think of them as either an enlargement or reduction of each other.

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.

Two triangles, one larger than the other. Each one has each side labelled with measurements.

Example 2:

The two triangles below are similar triangles as they have at least two corresponding angles that are equal in size. Both the large triangle and the smaller, purple, triangle share the \(\angle B\). And, as lines \(DE\) and \(AC\) are perpendicular, \(\angle A\) and \(\angle D\) are equal, and \(\angle C\) and \(\angle E\) are equal.

Two triangles, one smaller one sitting inside a large triangle. Each angle is labelled with a letter.

Example 3:

We can use these relationships to find unknown values of sides of triangles.

Given the similar triangles below, find the value of lines \(JL\) and \(KL\).

Two triangles, one smaller one sitting inside a large triangle. Each angle is labelled with a letter.

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 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 we can use to show us the relationship between the acute angles in a right-angled triangle and the length of its sides. You are likely to only need to use three of these functions. They are:

  • Sine (\(\sin\))
  • Cosine (\(\cos\))
  • Tangent (\(\tan\))

While using trigonometric functions, the angle you are using in the right-angled triangle is represented by the symbol \(\theta\), known as theta.

A right-angled triangle, with one angle labelled with the Greek letter theta. The right angle is marked with a box.

Each side of the right-angled triangle has a name. The longest side, that is always opposite the right angle, is known as the hypotenuse. The other two sides are called the opposite and the adjacent sides. The opposite side is always the side that sits opposite the angle we are using in our calculation. The adjacent side is the side that sits next to the angle, along with the hypotenuse.

A right-angled triangle. One of the angles is labelled with theta, the side opposite the angle is labelled opposite, the side next to theta is labelled adjacent, the longest side is labelled the hypotenuse.

We can use trigonometric functions to calculate the trigonometric ratios of an angle with respect to the sides of a right-angled triangle. To find the sine ratio of the angle, we divide the value of the side opposite to the angle by the value of the hypotenuse. To find the cosine ratio of the angle, we divide the value of the adjacent side to the angle by the hypotenuse. To find the tangent ratio of the angle, we divide the value of the side opposite the angle by the side adjacent to the angle.

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:

Table outlining the acronym SOHCAHTOA. S is for sine, c is for cos, t is for tan. And then each O is for opposite, a is for adjacent, and h is for hypotenuse.

Example: Given the triangle below, find the values of angles B and C to two decimal places.

A triangle, with angles labelled A, B and C, with A at the right angle. One side is labelled 12cm, the hypotenuse is labelled 15cm.

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 in any triangle where a side and its opposite angle are known.

When we’re given a triangle, for example:

A triangle with angles labelled A, B, and C in capital letters, and sides labelled a, b, and c in lower case letters. Each side is opposite it's respective angle, for example, angle A, is opposite side a, etc.

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.

An acute triangle, one side is labelled 21cm, the other is labelled K. Two of the angles have values, 85 degrees and 50 degrees

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.

An acute triangle, one side is labelled 42cm, the other is labelled 57cm. One angle is labelled 85 degrees, the other is labelled theta.

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 you understanding of the sine rule by working through the questions below.

Cosine rule

The cosine rule is used in any triangle when relating all three sides to one angle. Consider the triangle below:

A triangle with an angle labelled A, and sides labelled a, b, and c in lower case letters. Angle A is opposite side a.

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.

A scalene triangle with an angle labelled 120 degrees, and sides labelled a, 14cm and 8cm. Side a is opposite the known angle.

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.

A scalene triangle with an angle labelled theta, and sides labelled 14m, 13m and 12m. The unknown angle is opposite the 13m side.

Solution:

\[\begin{aligned} cos\;A &= \frac{b^2 + c^2 - a^2}{2bc} \\\ cos\;\theta &= \frac{12^2 + 14^2 - 13^2}{2 \times 12 \times 14} \\\ cos\;\theta &= \frac{12^2 + 14^2 - 13^2}{2 \times 12 \times 14} \\\ cos\;\theta &= \frac{171}{336} \\\ cos\;\theta &= 0.50893 \\\ \theta &= cos^{-1}\,0.50893 \\\ \theta &= 59.17^\circ \\\ \end{aligned}\]

Check your understanding by working through the questions below.

What formula should I use?

A flow chart leading you through how to choose a trigonometry formula.