Higher GCSE

# How to use triangles to remember exact sin, cos and tan values

If you’re taking the higher maths GCSE, there are certain values of sin, cos and tan that you need to know off by heart. Luckily in two of the three exams you will be allowed to use a calculator, but in the non-calculator exam you will need to know the following common values:

sin(0)                    cos(0)                    tan(0)
sin(30)                  cos(30)                  tan(30)
sin(45)                  cos(45)                  tan(45)
sin(60)                  cos(60)                  tan(60)
sin(90)                  cos(90)

In a previous blog, I showed you a method for finding the exact sin, cos and tan values using a table. In this blog, I will show you an alternative method.

The first thing to note about this method is that it can only be used to find these values:

sin(30)                  cos(30)                  tan(30)
sin(45)                  cos(45)                  tan(45)
sin(60)                  cos(60)                  tan(60)

So if you use this method you will need to remember these off by heart:

sin(0)                    cos(0)                    tan(0)
sin(90)                  cos(90)

This method will involve using Pythagoras’ theorem and trigonometry, so if you’re unsure on either of these I would recommend re-reading these blogs first:

A guide to Pythagoras’ theorem
A guide to Trigonometry – Part 1

## How to find sin(45), cos(45) and tan(45)

First of all, we need to draw an isosceles right-angled triangle. This will have angles of 90°, 45° and 45°.

Because it is an isosceles triangle, the horizontal and vertical sides will have the same length. Make this side length 1 (you can choose any number and it will still work, but 1 is the easiest).

We can use Pythagoras’ theorem to find the length of the hypotenuse (which I will call x).

We now know all the angles and sides of our triangle.

Now, we can use the trigonometric ratios (SOHCAHTOA).

It doesn’t matter which angle we focus on here, because they are both 45°. I’m going to choose the angle highlighted in red.

Label the Opposite (O), Adjacent (A) and Hypotenuse (H):

Now we can use SOHCAHTOA to find the sin, cos and tan values for 45°:

So there we are – we have worked out all three values, and we have practiced Pythagoras’ theorem and trigonometry in the process.

## How to find the 30 and 60 values

This will be a similar approach, but may require a bit more work. The benefit of this method is that once you have done the work, you can obtain all the 30 and 60 values.

First, start off with an equilateral triangle of side length 2.

Now we are going to split the triangle in half vertically, creating two congruent right-angled triangles. We are going to discard the left triangle and focus on the right one.

Because we split the equilateral triangle in half, the base of our right-angled triangle has length 1 and has two angles of size 30° and 60°.

Now, we have a missing side that we can use Pythagoras’ theorem to find (I will call this y).

Now that our right-angled triangle is fully labelled, similarly to the previous triangle we can use trigonometry to find the exact values.

Let’s do sin(30), cos(30) and tan(30) first, by focusing on the red angle.

Label the Opposite (O), Adjacent (A) and Hypotenuse (H):

Now we can use SOHCAHTOA to find the sin, cos and tan values for 30°:

Now we can find sin(60), cos(60) and tan(60) by focusing on the blue angle.

Re-label the Opposite (O), Adjacent (A) and Hypotenuse (H):

Now we can use SOHCAHTOA to find the sin, cos and tan values for 60°:

## Advantages and disadvantages of this method compared to the table method

The disadvantage of this method is that is can only find the 30, 45 and 60 values. The 0 and 90 values will need to be remembered off-by-heart.

The advantage of the method is it acts as practice of Pythagoras’ theorem and SOHCAHTOA. These are both integral parts of the foundation and higher tier syllabuses so are important to keep on top of.

Personally, I prefer the table method as technically there are fewer steps and fewer things to remember – and the table method will find all the values whereas this method will only find the 30, 45 and 60 values. But both methods are valid and it’s up to you which one you choose to use.

You can read a guide to the “table method” here.

I hope you found that useful. To practice using exact sin, cos and tan values in a range of GCSE-style questions, try our worksheet.

If you are in Bristol and need someone to work through this or any other aspect of maths with you, book a free taster session.

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