Two Triangles To Help You Master Exact Trigonometric Functions
GCSE Maths Level 4/5 topic
Within the GCSE maths exams (non-calculator), there are a number of trigonometric functions you need to remember. In particular, the following:
sin(0), sin(30), sin(45), sin(60), sin(90)
cos(0), cos(30), cos(45), cos(60), cos(90)
tan(0), tan(30), tan(45), tan(60), tan(90)This might seem like a lot of values to remember, however that’s where these two triangles can help you…
Triangle 1:
Let’s start with an equilateral triangle of side length two. We know all triangles have three angles which add to give us 180 degrees. Because all of the sides are the same length, that means all the angles are equal at 60 degrees each.
Bisecting (splitting) the triangle down the middle into two we are left with our first triangle. The bottom length is halved and therefore of side length 1, the top angle is also halved and becomes 45 degrees, however, the right hand side length is currently unknown.
The new triangle we’ve created is right-angled, meaning we can use Pythagoras’s Theorem to work out the remaining side length.
Pythagoras’s Theorem says that:
where c is the length of the hypotenuse – the triangles longest side.
Here we know a and c, meaning we are able to work out the remaining unknown side length, b.
Now we know all the side length values, we can look to work out the exact trigonometric values for the angles of the triangle.
Using SOH CAH TOA, we can work out the following trigonometric values:
sin(30), sin(60)
cos(30), cos(60)
tan(30), tan(60)
sin(angle) = opposite / hypotenuse
cos(angle) = adjacent / hypotenuse
tan(angle) = opposite / adjacent
Triangle 2:
For the second triangle, we will take an isosceles triangle of side length one. Again, it’s a right angled triangle and the remaining two angles are equal, at 45 degrees each.
Currently we don’t know the length of the hypotenuse, so let’s use Pythagorus’s Theorem again…
We can now use SOH CAH TOA again to work out:
sin(45)
cos(45)
tan(45)
sin(angle) = opposite / hypotenuse
cos(angle) = adjacent / hypotenuse
tan(angle) = opposite / adjacent
You will have noticed we haven’t yet covered the following exact trigonometric values:
sin(0), sin(90)
cos(0), cos(90)
tan(0), tan(90)We can find these values from looking at the graphs of each of the trigonometric functions:
From the above graphs, we can find the following values and therefore complete our set of exact trigonometric values.
sin(0) = 0
sin(90) = 1
cos(0) = 1
cos(90) = 0
tan(0) = 0
tan(90) = infinity