# Trigonometry Missing Sides

This chapter explores trigonometry missing sides. The objectives for this chapter are being able to use the sine, cosine and the tangent formulae to find the missing sides in a triangle. You may want to go through the ‘Trigonometry missing angles’ chapter before attempting this chapter.

## Important notes

In the ‘Trigonometry missing angles’ we explored the trigonometry formulae and came up with SOHCAHTOA to help in remembering them. The concept explored here is very similar to what we explored in the missing angles chapter. We saw that given the triangle below;

The formulae that can be used to find the angle a° are;

## Finding missing sides using sine

This section explores finding missing sides using cosine. Below is a triangle;

First we identify the sides that are known or available. We have the Adjacent and Hypotenuse x is the adjacent and 13 is the hypotenuse. Now we need to identify what function to use whether sine, cosine, or tangent. We could use SOHCAHTOA to help remember the full formula;

Above we can see that we would need to use the cosine function, so the formula to use is;

Next we substitute in the known values as shown below;

…we rearrange to get x on its own since it is the value that we are after.

Now we need to work out the value of 13cos(40°) on the calculator and round off to 2d.p, the answer would be;

## Finding missing sides using sine

This section explores finding missing sides using cosine. The triangle below will be used as an example.

First we identify the known sides. We know the Opposite and Hypotenuse; x is opposite while 41 is the hypotenuse. Next we need to identify the function to use; whether sine, cosine, or tangent.

Above we can see that we need to use the sine function. The formula that we need to use is shown below.

Then we substituted in the values in the formula.

…next we rearrange to get x on its own since it is the value that we’re trying to find;

Next we work out 41sin25° calculator and round off to 2 decimal places.

## Finding missing sides using tangent

This section explores finding missing sides using tangent. The triangle shown below should help in this practice.

First we need to identify the known or available sides. We know the Opposite and Adjacent; x is opposite while 24 is the adjacent. Next we need to identify the function to use;

The function to use is the tangent formula as we can see above the function states that;

Next we substitute in the known values as shown below;

…then we rearrange to get x on its own since it is the value we’re trying to find;

Next we work out 24tan13° on the calculator and round off to 2 decimal places.

## x on the bottom of the equation

You’ll find some difficult when x is on the bottom of the equation. The triangle below should help to explore the problem;

First we identify the known or available sides. In the triangle above we know the Adjacent and Hypotenuse. 17 is adjacent and x is the hypotenuse, looking at;

…we can see that the function to use is cosine. The following is the formula in connection with cosine;

Next we substitute in the values;

In this instance the x is at the bottom of the equation this makes it much harder to rearrange to make x the subject unless you have encountered this form of rearranging before. First multiply both sides by x or cross multiply as shown below.

…that gives;

…next we divide both sides by cos50° to result in;

Easy isn’t it? Next work out 17÷cos50° and round off to 2 decimal places to get the result;

## Exam question

This section looks at a previous exam question;

- Calculate the length x
- Calculate the angle CDB

- Let angle CDB = a