Find the values of 𝑎 and 𝑏.
Looking at the diagram, we can see that we have a right-angled triangle, in which the other two angles are 30 degrees and 60 degrees. The hypotenuse of the triangle, that is, the longest side, has been given; it’s 12 units. And we’re asked to find the values of 𝑎 and 𝑏, which are the lengths of the other two sides.
When answering questions about right-angled triangles, there are two approaches that spring to mind: the Pythagorean theorem and right-angled trigonometry. The Pythagorean theorem, remember, tells us about the relationship between the lengths of the three sides of the triangle. And therefore, we apply it when we know the lengths of two sides. As we’ve already been given the length of one side in this triangle, we can’t apply the Pythagorean theorem.
Trigonometry, however, tells us about the relationship that exists between the lengths of sides and the sizes of angles in right-angled triangles. As we’ve been given the length of a side and the sizes of the angles, we can apply right-angled trigonometry to this problem.
First let’s recall the acronym SOHCAHTOA, which tells us which of the three trigonometric ratios — sine, cosine, or tan — we should be using, depending on which pair of sides we’ve been given. Let’s look at how to calculate side 𝑎 first of all.
We’ve been given the sizes of both of the non-right angles. But we need to choose one of them to work with. So I’m going to choose the angle of 30 degrees. I’ll begin by labelling the three sides of the triangle in relation to this angle of 30 degrees. The hypotenuse is always the side directly opposite the right angle. So that’s the side of length 12. The opposite is the side opposite the given angle. So in the case of the angle of 30 degrees, this is the side 𝑎. The adjacent is the third side, which is always between the known angle and the right angle.
Now we see that side 𝑎 is the opposite, and the side we know is the hypotenuse. O and H appear together in the SOH part of SOHCAHTOA. So this tells us that it’s the sine ratio we need to use in order to calculate side 𝑎.
Let’s recall its definition. Sin of an angle 𝜃 is equal to the opposite divided by the hypotenuse. This ratio would always be the same for an angle of size 𝜃, no matter how big the triangle is. Substituting the values in this question — so 𝜃 is 30 degrees, the opposite is 𝑎, and the hypotenuse is 12 — we have the equation sin 30 degrees is equal to 𝑎 over 12.
Now here’s a really key fact. 30 degrees is a special angle, for which the trigonometric ratios sine, cosine, and tangent can all be expressed quite simply in terms of fractions or surds. In fact, sin of 30 degrees is equal to a half. The ratio of the opposite divided by the hypotenuse is always one over two if the angle is 30 degrees. This means then that we have the relatively straightforward equation 𝑎 over 12 is equal to a half, and we can solve for 𝑎.
To solve, we multiply both sides of the equation by 12, giving 𝑎 is equal to 12 multiplied by a half, which is six. So by remembering that the ratio between the opposite and the hypotenuse is always equal to one-half if the angle in question is 30 degrees, we’ve found the value of 𝑎.
Now let’s think about how to find the value of 𝑏. There are a number of different approaches that we could take. We now know two sides of the right-angled triangle. So we could apply the Pythagorean theorem to calculate 𝑏 if we wanted. However, let’s continue with our approach using trigonometry.
If we look at the ratio between side 𝑏 and the side of length 12, this is the ratio involving the adjacent and the hypotenuse. A and H appear together in the CAH part of SOHCAHTOA, which tells us that it’s the cosine ratio we’re interested in. Its definition is that cos of the angle 𝜃 is equal to the adjacent divided by the hypotenuse. Substituting 30 degrees for the angle, 𝑏 for the adjacent, and 12 for the hypotenuse, we have the equation cos of 30 degrees is equal to 𝑏 over 12.
Now again, we have a key fact about the cos ratio for 30. The ratio between the adjacent and the hypotenuse is always equal to the square root of three over two if the angle in question is 30 degrees. Substituting this into the equation gives 𝑏 over 12 is equal to the square root of three over two. And we can solve this equation to find the value of 𝑏.
Multiplying both sides by 12 gives 𝑏 is equal to 12 root three over two. This can be simplified slightly by cancelling a factor of two from both the numerator and the denominator. The value of 𝑏 is six root three.
Now we answered this question using the angle of 30 degrees. But an equally valid approach would have been to use the angle of 60 degrees. Let’s see what difference that would have made. In relation to the angle of 60 degrees, the opposite and the adjacent sides would be the other way round. So 𝑏 would be the opposite, and 𝑎 would be the adjacent.
When we looked at calculating side 𝑎, the two sides involved in the ratio would be the adjacent and the hypotenuse. So we’d be using the cosine ratio. Instead of the equation sin 30 degrees is equal to 𝑎 over 12, instead we’d have the equation cos of 60 degrees is equal to 𝑎 over 12. However, this wouldn’t make any difference to our answer as sin of 30 degrees and cos of 60 degrees are both equal to one-half.
In the same way, when calculating the length of side 𝑏, the two sides involved in the ratio will be the opposite and the hypotenuse, which means we’ll be using the sine ratio. So instead of cos 30 degrees equals 𝑏 over 12, we’d have that sin of 60 degrees is equal to 𝑏 over 12. However, just like cos of 30 degrees, sin of 60 degrees is also equal to root three over two. So our calculation for 𝑏 would be the same.
You can answer this question in relation to the angle of 30 degrees or in relation to the angle of 60 degrees or indeed a mixture of the two. And it will give the same answer. 𝑎 is equal to six. 𝑏 is equal to six root three.