Trigonometry - Finding Unknown Sides

Continuing with trigonometry, we looked at finding unknown side lengths. To start with, we need to familiarise ourselves with the following acronym:

SOH CAH TOA

This acronym stands for Sin, Opposite, Hypotenuse, Cos, Adjacent, Hypotenuse, Tan, Opposite, Adjacent. We use this to help us to remember what each of the trig ratios is (i.e. Sin is Opposite over Hypotenuse, which we can get from the order of the acronym). 

When we are finding side lengths, we want to rearrange this acronym into sets, with a set for each trig ratio. The best bet is to arrange them into triangles like so:

We can see that if we move left to right through these triangles we still have our acronym in its correct order. Now you are probably all wondering "why do we need to do this?". All will be revealed soon. 

Now when we are faced with a problem like so:

           Find the unknown value in the following problem

To solve problems like this one, we need to use 5 key steps. They are:

1. Label the sides (i.e. label the hypotenuse, the adjacent and the opposite sides)
2. Find which sides we are using for this problem (i.e. which sides are we given some information about)
3. Find which ratio requires us to use these sides (i.e. if we are using the opposite and the hypotenuse, then we will have to use the Sin ratio)
4. Write out the equation required to solve the problem
5. Substitute in our values and solve our equation. 

So using these steps, lets solve the problem above. 

Firstly we have to label our sides. If you are unfamiliar with how to do this, look at the posts below. When we do this for our example above we get: 

Now step two is to find which sides we are actually using for this problem. If we look at the triangle, we are given the length of the hypotenuse. We are also given an unknown value for the opposite side. We are not given any information regarding the adjacent side, so we won't be using this side. For this triangle the two sides we are using are the opposite side and the hypotenuse. 

Step 3 is to find which ratio uses these two particular sides. In this triangle we know it will be the Sin ratio, because the two sides we are using are the opposite side and the hypotenuse. 

Step 4 is to write out the equation. To do this we need to use the triangles we looked at above. Since we know we are using the Sin ratio (from step 3), we only need to look at the first of the 3 little triangles: 

We can use this triangle to help us write the equation. Since we are trying to find the value of the opposite side, we know we want the equation to be O = "something"

To work our what this "something" is, we use the triangle. Since we are finding the value for O (the opposite side), we cover this up in our triangle. We are now left with:

The "S" stands for "sin of an angle" and the "H" stands for "the length of the hypotenuse". Since they are next to one another, we need to multiply them together. This is the "something" in our equation. So finally, for this step, our equation is:

O = sin Ө x H

Step 5 is to substitute our values into this equation and solve it. For this triangle, our value for Ө is 40 (the size of the angle in degrees) and our value for H is 64cm (the length of the hypotenuse). So to solve this equation for O, we substitute these values into our equation and then use a calculator to solve it. 

When we substitute our values in we get:

O = sin (40) x 64

Putting this into our calculator we get a value of 41.1cm  to 1 decimal place for our opposite side. 

Looking at one more example, we may have: 

             Find the unknown for the following triangle:

Again, following the same 5 steps, we can easily solve this one. 

Step 1: Label the sides

Step 2: We will be using the opposite and the adjacent sides

Step 3: This means that we will be using a Tan ratio. 

Step 4: We will be using the following triangle:

We don't know our adjacent side, so we want the equation to be A = "something". To work out this "something" we cover up the part we want to find out, so in this case the A. 

We are left with O over T, or to be more specific "the length of our opposite side" over "tan of our angle". Now because they aren't next to one another like the previous example, we do not multiply. This time, they are over one another, so we are dividing. This gives us a final equation of:

A = O/tan Ө 

Step 5 gives us:

A = 38.2/tan (27)

If we put this into our calculator we get a final answer of 75.0cm to 1 decimal place for our adjacent side

This should give you enough information to work through the 4th set of problems on the worksheet (which can be found in the worksheets tab). If you have any questions please feel free to comment on here or send me an email.