Video Transcript
Find the exact value of sin of 30 degrees.
Now, you may be used to using a calculator or a computer when working with
trigonometric functions. This is because, for most angle values, sin, cos, and tan evaluate to a number with
many decimal places. In certain cases, however, sin, cos, and tan evaluate to a whole number or a number
that can be written as a fraction. The common cases that you may come across are for zero degrees, 30 degrees, 45
degrees, 60 degrees, and 90 degrees.
Looking at our question, we can see that we do indeed have one of these cases. Although trigonometric functions are closely related to right-angled triangles, in
this case, we’re going to start by looking at an equilateral. We know that, for any equilateral triangle, all angles are 60 degrees and all sides
are equal. We could choose any side length for this triangle and get the correct answer. However, we’re going to choose two to make things simpler later on.
The first thing we’re going to do to our triangle is to chop it in half with a line
perpendicular to the base. We can now discard one of the halves of our triangle. In doing this, we’re left with a new right-angled triangle. Since both the top angle and the base have been bisected, we know that they are half
of their original values. This leaves us with a top angle of 30 degrees and a base of one. Let’s label the sides of our new triangle as 𝑎, 𝑏, and 𝑐.
To complete the information for this triangle, we need to find side length 𝑎, and we
can do so using the Pythagorean theorem. The theorem states that, for any right-angled triangle, the square of the hypotenuse
is equal to the sum of the square of the other two sides.
Let’s substitute in the values that we have for 𝑏 and 𝑐. Squaring our numbers, and we can see that 𝑎 squared plus one equals four. Subtracting one from both sides, we can see that 𝑎 squared equals three. Finally, taking the square root of both sides, we can see that 𝑎 is equal to the
square root of three.
It’s worth noting that there are actually two solutions to this equation. We could say that 𝑎 is equal to the positive square root of three or the negative
square root of three. In our case, 𝑎 is a length, and therefore we’re only interested in the positive
solution.
Now that we have found our answer, we have a completed right-angled triangle. The triangle can help us to solve this kind of problem by using the trigonometric
identity for sine. The identity tells us that, for any right-angled triangle, sin of an angle 𝜃 is
equal to the length of the side opposite that angle divided by the length of the
hypotenuse.
We can see from our triangle, that we do indeed have an angle of 30 degrees. We can therefore write that sin of 30 degrees is equal to the length of its opposite
side — that’s 𝑏 — divided by the length of the hypotenuse — that’s 𝑐. Since we have these values, we can substitute one for 𝑏 and two for 𝑐, which gives
us that sin of 30 degrees is equal to one over two, or a half.
Now you may be wondering why we found the value for side length 𝑎 since this was not
involved in the calculation of sin 30 degrees. The reason is that this triangle is very useful in the calculation of some of the
other exact trigonometric ratios.
As a quick aside, let’s consider the angle 60 degrees in this triangle. We can see from our sine identity that the sin of 60 degrees is equal to the length
of its opposite side — that’s 𝑎 — over the length of the hypotenuse — that’s
𝑐.
Again, we have these values in our triangle, and so we can substitute in the square
root of three for 𝑎 and two for 𝑐. On doing so, we find that the sin of 60 degrees is equal to the square of three over
two. As a final note, this triangle can also help you answer similar questions. By using this method and recalling the other trigonometric identities, you should be
able to answer questions involving the exact values for cos of 30 and 60 degrees and
tan of 30 and 60 degrees.
