Trigonometric Function Sine

In this article we will discuss trigonometric functions. The word trigonometric comes from Greek words trigónon which means triangle and metrein which means measure.

Example

Suppose we have a triangle and we know the length of one leg and we know the size of an angle. See example:

The right-angeled triangle <i>ABC</i>

The |AC| = 3 expression means that the side AC has length equals to three. It could be three inches or three football stadiums it doesn’t matter. The question is: can we somehow compute the length of the other two sides? If you remember the Pythagorean theorem you know that you can compute the length of one side if you know the lengths of the others. But in this case we know just the length of one side. How unfortunate.

But worry not! We can use the power of trigonometric functions and we can compute the lengths we don’t know using the size of the angle we know. But we need to be able to distinguish those two triangle’s legs. Let’s give them names:

  • The side AB is adjacent to the angle α,
  • the side AC is opposite to the angle α.

See updated image:

The adjacent side and the opposite side

Now we can use the sine function. The definition follows: For the angle α, the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse. In our case, this holds:

\Large\sin(\alpha)=\frac{\mbox{length of opposite side}}{\mbox{length of hypotenuse}}=\frac{|AC|}{|BC|}

So the input of the sine function is the size of an angle. You can use either degrees or radians, it doesn’t matter. The output is the ratio of the length of the opposite side to the length of the hypotenuse. This output is always a number from interval (0, 1).

How to compute the length using the sine function

Look at the equation again. We know the left side, the sin(α), we know the length of AC and the only part that we don’t know is the length of BC. The value of sin(α) is just a constant number, you can calculate it using your calculator or using e. g. Google). As you can see, the value of sin(30°) equals to 0.5. Now we can adjust the equation such that on the left side is just the only unknown variable:

\begin{eqnarray}\sin(\alpha)&=&\frac{|AC|}{|BC|}\qquad /\cdot |BC|\\|BC|\cdot\sin(\alpha)&=&|AC|\qquad /\cdot\frac{1}{\sin(\alpha)}\\|BC|&=&\frac{|AC|}{\sin{\alpha}}\\\end{eqnarray}

Now we can substitute the right side of the equation:

|BC|=\frac{|AC|}{\sin{\alpha}}=\frac{3}{0.5}=6

So we computed that the length of the side BC is equals to 6. We can add the result to the image we have:

The triangle <i>ABC</i>

Now we know the length of the two sides so we can simply use Pythagorean theorem to compute the length of the last one … or we can use the cosine function! Let’s do this because it’s gonna be more fun and, well, this article is called trigonometric functions, not Pythagorean theorem, right?