Trigonometric Ratios

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Have you ever looked down at your calculator and wondered what certain buttons actually do? For example, you may have seen the buttons ‘sin’, ‘cos’ and ‘tan’ but have never known what they actually mean. These three buttons are essential to our study of Trigonometry and in this article, we will learn all about what they mean, and how we use them to answer questions. However, before we begin, we must first recap Pythagoras' Theorem to provide a bit more of a background.

Pythagoras and Trigonometry Differences

By now, you have probably already studied Pythagoras’ theorem. In this section, we will quickly recap what we mean by Pythagoras' theorem and establish the key differences between Pythagoras and Trigonometry.

Trigonometric Ratios, Triangle set up for Pythagoras' Theorem, Jordan Madge Pythagoras- triangle set up for Pythagoras' theorem, Jordan Madge- StudySmarter Originals

For any given right-angled triangle, the longest side of the triangle is called the hypotenuse. The hypotenuse is the side that appears slanted and is always the side opposite the right angle.

Pythagoras' Theorem

Pythagoras’ theorem states that for any given right-angled triangle, with hypotenuse, c, and the other two sides labelled a and b (as depicted in Figure 1), $a^{2} + b^{2} = c^{2}$ .

Note: a and b in figure 1 are interchangeable, however, c must always be the hypotenuse.

Suppose we have the below triangle, find the side labelled x.

Pythagoras' theorem, Example finding hypotenuse, Jordan Madge Pythagoras- Example Finding Hypotenuse, Jordan Madge- StudySmarter Originals

Solution:

First, let's label each of the sides a, b and c:

Pythagoras- Example Finding Hypotenuse, Jordan Madge- StudySmarter Originals

Now, Pythagoras' theorem states that $a^{2} + b^{2} = c^{2}$ .

Substituting $a = 3, b = 4, c = x$ into this, we obtain $9 + 16 = x^{2}$ , or $x^{2} = 25$ .

Therefore, to find x we simply take the square root both sides. Thus, $x = 5 c m$ .

In the above example, we have what is known as a Pythagorean Triple. This is where each of the three sides of a right-angled triangle is an integer.

Suppose we have the below triangle, find the side labelled x.

Pythagoras' theorem, Example finding missing side, Jordan Madge

Pythagoras- Example Finding Missing Side, Jordan Madge- StudySmarter Originals

Solution:

First, let's label each of the sides a, b and c:

Pythagoras' theorem, Example finding missing side, Jordan Madge

Pythagoras- Example Finding Missing Side, Jordan Madge- StudySmarter Originals

Now, this example is slightly different from the previous as this time we are not finding the hypotenuse, c. Therefore, we have to use a rearranged version of Pythagoras' theorem: $b^{2} = c^{2} - a^{2}$ . Substituting $a = 2, b = x$ and $c = 5$ into this formula, we obtain $x^{2} = 5^{2} - 2^{2} = 25 - 4 = 21$ . Thus, we simply need to square root both sides to obtain that $x = \sqrt{21} = 4.58 c m$ (2.d.p).

Now, the key thing to note with Pythagoras's theorem is that it only works for right-angled triangles where we are given two sides and wish to find the third side. It doesn't involve any Angles. However, what if we wanted to find a missing angle? What if we were given an angle but not enough sides? This is where trigonometry comes in!

The purpose of trigonometry is to find missing lengths and Angles. For the time being, we will simply consider trigonometry in right-angled triangles. However, later on in another article, we will consider trigonometry in triangles that are not necessarily right-angled using what is known as the sine and cosine rule.

Trigonometric Ratios

Trigonometric Ratios, Triangle set up for Trigonometry, Jordan Madge Trigonometry- Right-angled triangle with Opposite, Hypotenuse and Adjacent sides labelled, Jordan Madge- StudySmarter Originals

Now, let's dive straight into trigonometry. Like with Pythagoras, we will start by defining our triangle and labelling some key properties. Above depicted in figure 2 is a right-angled triangle with an angle labelled with the Greek symbol theta ( $θ$ ). For some reason, mathematicians just like using this symbol to denote missing angles. Now, as before with Pythagoras, the hypotenuse is the longest side opposite the right angle. We now introduce two new labels for the other sides: the adjacent side and the opposite side.

The opposite side of a right-angle triangle is the side opposite the angle $θ$ and we usually label it O. The adjacent side, is the side adjacent (next to) to the angle $θ$ that is not the hypotenuse. We usually label this A.

Trigonometric Ratios Definition

Now that we have set up our triangle, we can define our trigonometric ratios. This is where sin, cos and tan come in handy!

Trigonometric Ratios, Right Angled Triangle with O, A and H labelled, Jordan Madge Trigonometry- Right-angled triangle with opposite (O), adjacent (A) and hypotenuse (H) sides labelled, Jordan Madge- StudySmarter Originals

Trigonometric Ratios Formulas

Given the right-angled triangle in figure 3 with hypotenuse H, opposite side O and adjacent side A, we have the following ratios:

$\sin (θ) = \frac{O}{H} \cos (θ) = \frac{A}{H} \tan (θ) = \frac{O}{A}$

It may be worth noting at this point that sin is short for sine, cos is short for cosine and tan is short for tangent.

These ratios can be remembered using the acronym SOHCAHTOA. Although, learning to spell that acronym may be a challenge in itself. Now, at this point, it is okay if you are feeling a little bit lost or confused. It will all become clear in a few short moments. For now, we just need to accept that these ratios exist and that they are very useful.

Trigonometric Ratios Examples

Examples Involving Missing Lengths

So far we have covered all of the tools that you will need to answer questions involving trigonometric ratios. However, to really understand what it all actually means, we must go over some examples. Note, that the purpose of this entire exercise is to find missing lengths or angles. In this section, we will focus on finding missing lengths.

Steps to find the missing side

To find the missing side of a right-angled triangle using trigonometry, there are a few steps.

Step 1: Label the sides O, A and H.

Step 2: Work out which sides are involved. In other words, which sides do we either know or want to know?

Step 3: Identify the relevant trigonometric Ratio.

Step 4: Set up the appropriate equation.

Step 5: Solve the equation to find the missing side.

Suppose we have the below triangle. Find the side labelled x.

Trigonometry- Example Finding Missing Side, Jordan Madge- StudySmarter Originals

Solution:

First, let's label each of the sides O, A and H.

trigonometry, Example finding missing side, Jordan Madge

Trigonometry- Example Finding Missing Side, Jordan Madge- StudySmarter Originals

The next step is to work out which two of the three sides are involved. What we mean by this, is which sides do we know or wish to know. In this example, we know the hypotenuse is $3 c m$ and we want to know the adjacent side, $x c m$ . Thus, the two sides involved are the adjacent and hypotenuse. Next, we must identify which trigonometric Ratio involves the adjacent and hypotenuse. In this case, it is cos, because cos is the only identity that involved both the adjacent and hypotenuse side. Substituting $θ = 42, A = x, H = 3$ into $\cos (θ) = \frac{A}{H}$ we obtain $\cos (42) = \frac{x}{3}$ .

All we have to do now is a bit of rearranging to find x. We can do this by multiplying both sides by 3. Thus, $x = 3 \cos (42)$ . We get the final answer by typing this into our calculator. So, $x = 2.23 c m .$

Suppose we have the below triangle. Find the side labelled x.

Trigonometry- Example Finding Missing Side, Jordan Madge- StudySmarter Originals

Solution:

First, let's label each of the sides O, A and H.

trigonometry, Example finding missing side, Jordan Madge

Trigonometry- Example Finding Missing Side, Jordan Madge- StudySmarter Originals

In this case, we know the hypotenuse, and we wish to know the opposite side. Therefore we use sin since sine is the only trigonometric ratio involving the hypotenuse and opposite side.

Now,

\sin (θ) = \frac{O}{A}

\sin (27) = \frac{x}{3.7}

. Multiplying both sides by 3.7, we get

x = 3.7 \sin (27)

which is 1.68 cm (3.s.f).

Suppose we have the below triangle. Find the side labelled x.

Trigonometry- Example Finding Missing Side, Jordan Madge- StudySmarter Originals

Solution:

First, let's label each of the sides O, A and H.

trigonometry, Example finding missing side, Jordan Madge

Trigonometry- Example Finding Missing Side, Jordan Madge- StudySmarter Originals

In this case, we know the adjacent side, and we wish to know the hypotenuse. Therefore we use cos.

Since $\cos (θ) = \frac{A}{H}$ we have $\cos (51) = \frac{8}{x}$ .

Now, to rearrange this, we need to first multiply both sides by x to obtain $x \cos (51) = 8$ . Then, to find x, we divide both sides by $\cos (51)$ to get $x = \frac{8}{\cos (51)} = 12.7 c m$ (3.s.f).

Suppose we have the below triangle. Find the side labelled x.

Trigonometry- Example Finding Missing Side, Jordan Madge- StudySmarter Originals

Solution:

First, let's label each of the sides O, A and H.

trigonometry, Example finding missing side, Jordan Madge

Trigonometry- Example Finding Missing Side, Jordan Madge- StudySmarter Originals

In this case, we know the opposite side, and we wish to know the adjacent side. Therefore we use tan.

Since $\tan (θ) = \frac{O}{A}$ we have that $\tan (30) = \frac{3}{x}$ . Multiplying both sides by x, we have $x \tan (30) = 3$ . Dividing both sides by $\tan (30)$ we get $x = \frac{3}{\tan (30)} = 5.20 c m$ (3.s.f).

Examples Involving Missing Angles

Inverse Trig Functions

To find missing angles using trigonometry, the steps are very similar to before. However we need to use inverse Trigonometric Functions. On your calculator, you may see $\sin^{- 1}, \cos^{- 1}, \tan^{- 1}$ above each of sin, cos and tan. You can find them by pressing the shift button and then the relevant trig function.

Using your calculator, find $\sin^{- 1} (0.1), \cos^{- 1} (0.1)$ and $\tan^{- 1} (0.1)$ .

Solution:

$\sin^{- 1} (0.1) = s h i f t + \sin (0.1) = 5 . 74^{°}$ (3.s.f)

$\cos^{- 1} (0.1) = s h i f t + \cos (0.1) = 84 . 3^{°}$ (3.s.f)

$\tan^{- 1} (0.1) = s h i f t + \tan (0.1) = 5 . 71^{°}$ (3.s.f)

Suppose we have the below triangle. Find the angle labelled $θ$ .

Trigonometry- Example Finding Missing Angle, Jordan Madge- StudySmarter Originals

Solution:

To find missing angles, the steps are pretty much the same as before. However, there is one minor difference. As before, let's start by labelling each of the sides O, A and H.

trigonometry, Example finding missing angle, Jordan Madge

Trigonometry- Example Finding Missing Angle, Jordan Madge- StudySmarter Originals

Now, we again need to identify which sides are involved. In this case, we know the adjacent and hypotenuse. Since cos involves adjacent and hypotenuse, we use the cosine.

Since $\cos (θ) = \frac{A}{H}$ , we have $\cos (θ) = \frac{1.2}{3}$ .

Now this time, to get theta by itself, we must take the inverse cosine of both sides.

Therefore, our answer is $θ = \cos^{- 1} (\frac{1.2}{3}) = 66 . 4^{°}$ (3.s.f).

Suppose we have the below triangle. Find the angle labelled $θ$ .

Trigonometry- Example Finding Missing Angle, Jordan Madge- StudySmarter Originals

Solution:

Labelling the sides, we can see that we have the opposite side and hypotenuse. Therefore we use sine.

trigonometry, Example finding missing angle, Jordan Madge

Trigonometry- Example Finding Missing Angle, Jordan Madge- StudySmarter Originals

Since $\sin (θ) = \frac{O}{H}$ , we have $\sin (θ) = \frac{2.8}{5.2}$

To get $θ$ by itself, we must take the inverse sine of both sides. Thus, $θ = \sin^{- 1} (\frac{2.8}{5.2}) = 32 . 6^{°}$ (3.s.f).

Trigonometric Ratios Table

For obvious reasons, trigonometry is a topic usually covered in calculator exams. However, there are some values of sin, cos and tan that you may be expected to know for your GCSE non-calculator exam. It's mean, I know. However, you should try your best to memorise these results.

Angle( $θ$ )	Sin( $θ$ )	Cos( $θ$ )	Tan( $θ$ )
30	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$
45	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	1
60	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$

Suppose we have the below triangle. Find the side labelled x.

(non-calculator question)

Trigonometry- Example Finding Missing Side, Jordan Madge- StudySmarter Originals

Solution:

Since we want to know the opposite side and we have the opposite side, we will use sine.

Since $\sin (θ) = \frac{O}{H}$ , we have $\sin (30) = \frac{x}{2}$ .

Rearranging, we find that $x = 2 \sin (30)$ .

Using table 1, we see that $\sin (30) = \frac{1}{2}$

Thus, $x = 2 \times \frac{1}{2} = 1 c m .$

Suppose we have the below triangle. Find the angle labelled $θ$ .

(non-calculator question)

trigonometry, Example finding missing angle, Jordan Madge

Trigonometry- Example Finding Missing Angle, Jordan Madge- StudySmarter Originals

Solution:

Since we have been given both the opposite and adjacent sides, we use tan.

Since $\tan (θ) = \frac{O}{A}$ , we have that $\tan (θ) = \frac{6.3}{6.3} = 1$ .

To get $θ$ by itself, we must take the inverse tan of both sides.

Thus, $θ = \tan^{- 1} (1)$

Using table 1, we see that $\tan (45) = 1$ and thus $θ = 45^{°}$ .

Trigonometric Ratios - Key takeaways

Trigonometric ratios are used when finding missing sides and angles in right angled triangles.
Trigonometry differs to Pythagoras as it involves angles.
We can use the acronym SOHCAHTOA to remember the trigonometric ratios.
For a given right angled triangle, we can label the hypotenuse and opposite and adjacent sides.
The Greek letter $θ$ is often used to denote angles.
We can use inverse trig Functions to find angles.

Frequently Asked Questions about Trigonometric Ratios

Trigonometric ratios are the ratios that we can use to help find missing sides or angles of a right angled triangle.

Sine, cosine and tangent are the names of the three trigonometric ratios. They can be remembered using the acronym SOHCAHTOA, where Sin(x)=O/H, Cos(x)=A/H and tan(x)=O/A

tan(x)= opposite/ adjacent

Hipparchus was been credited as "the father of trigonometry", and so one could say that he was the inventor of trigonometric ratios.

It depends what trigonometric ratio you are trying to find. If you are interested in working out the sine of an angle, you take the side opposite the angle and divide it by the hypotenuse. If you are interested in working out the cosine of an angle, you take the side adjacent to the angle and divide it by the hypotenuse. If you are interested in working out the tangent of an angle, you take the side opposite the angle and divide it by the side adjacent.

Flashcards in Trigonometric Ratios9 Start learning

When do we use inverse trig functions?

When finding angles.

What is the hypotenuse of a right angled triangle?

The longest side. It is always the side opposite the right angle.

What is the opposite side of a right angled triangle?

It is the side directly opposite the angle, labelled O.

What is the adjacent side of a right angled triangle?

It is the side adjacent to (next to) the angle that is not the hypotenuse.

How does Pythagoras theorem differ to trigonometry?

We use Pythagoras Theorem when we have two sides and wish to find a third. We use Trigonometry when we have either two sides and a missing angle or one side and one angle and we wish to find a missing side.

True or false: Trigonometry works for only right-angled triangles.

False. However, when using trigonometric ratios, we must have a right-angled triangle. Trigonometry can actually be used in triangles that are not right-angled using the sine and cosine rules but these are covered in another article.

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