4.10 F) Finding an Angle
Whenever we are finding what an unknown angle is using trigonometry, we use the inverse of the trigonometry function in the formula triangle. The inverse of sin is sin-1, the inverse of cos is cos-1 and the inverse of tan is tan-1. We use the inverse to get out of the trigonometric function. On most calculators, we get the inverse function by pressing shift and then the trigonometry function that we want to use; for example, we will get sin-1 by pressing shift and then sin.
The formula triangles are shown below.
Example 1
What is the size of angle x in the triangle below? Give your answer to the nearest whole number.
We start questions like this in the same way as when we were finding a length. The first step is to label all of the sides that we have or are trying to find. After labelling the sides, we can see what formula triangle we need to use. For the triangle above, we are given the opposite and adjacent. The labelled triangle is shown below.
We have the opposite and adjacent in the triangle. This means that we are going to be using the TOA formula triangle.
We are looking for the angle, which means that we cover up T in the TOA formula triangle, which tells us that we need to divide the opposite by the adjacent.
We can now sub the values in:
We have tan(x), but we want to find the value of x and not tan(x). To find the value of x, we need to get out of tan, which we do by doing the inverse to tan for both sides; the inverse of tan is tan-1. The tan and tan-1 on the left side cancel one another out, which means that we are left with just x.
The question asks us to give our answer to the nearest whole number.
Therefore, the answer to this question is 32°.
Example 2
What is the value of y in the triangle below? Give you answer to 1 decimal place.
Let’s start by labelling the sides that we are given in the question. We are given the opposite and the hypotenuse. The labelled triangle is shown below.
We have the opposite and the hypotenuse, which means that we will be using the SOH formula triangle.
We are looking for the angle, which means that we cover up S. This tells us that we find S by dividing the opposite by the hypotenuse.
We can now sub the values in.
We want to find out what y is and not sin(y). Therefore, we need to take the inverse of sin for both sides; the inverse of sin is sin-1. The sin and sin-1 on the left side of the equation cancel out, which leaves y.
We are asked to give the answer to 1 decimal place. We do this by placing a line after the first decimal place and looking at the number to the right of the line.
The number to the right of the line is 1, which is less than 5, so we round down. Therefore, y to one decimal place is 41.8°.