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How to find an unknown adjacent side or hypotenuse using cosine ratios.

 This topic is part of the TCS FREE high school mathematics 'How-to Library'. It shows you how to find an unknown adjacent side or hypotenuse in a right triangle using cosine ratios. (See the index page for a list of all available topics in the library.)Â To make best use of this topic, you need to download the Maths Helper Plus software. Click here for instructions.

### Theory:

The cosine ratio can be used to calculate the length of one of the shorter sides (legs) of a right triangle, or to find the length of the hypotenuse. In both cases you need to know the cosine ratio for one of the smaller angles. If you are finding an unknown leg length, then the hypotenuse must be known. If you are finding the hypotenuse, then one leg of the triangle must be known. The steps for both problem types are shown below:

Problem type 1: Finding the adjacent side.

Â Example 1: Use the cosine ratio to find the unknown adjacent side ‘x’ in this triangle:

Â Solution:

Â But cos60ºÂ  = 0.5Â  (From a table of cosine ratios, or scientific calculator with a ‘cos’ button.)

Â so xÂ  =Â  20 × 0.5

Â Â Â Â Â Â Â Â Â  =Â  10

Problem type 2: Finding the hypotenuse.

Â Example 2: Use the cosine ratio to find the hypotenuse in this triangle:

Solution:

Â But cos 35ºÂ  = 0.8192Â  (From a table of cosine ratios, or scientific calculator with a ‘cos’ button.)

### Method:

Maths Helper Plus shows you to how to solve triangles for an unknown adjacent side or hypotenuse by using cosine ratios. Full working steps and a labelled diagram are created. The steps below will show you how...

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#### Step 2Â  Display the parameters box

Press the F5 key to display the parameters box:

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You enter the given information into these edit boxes as follows:

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Â Â Â Â Â Â Â  edit box 'A' = given angle less than 90º.

NOTE: Just type a number for the angle A. Do NOT use the degree operator: ° after the angle.

Â Â Â Â Â Â Â  edit box 'B' = adjacent side to A.Â

Â Â Â Â Â Â Â  edit box 'C' = hypotenuse.

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Set one only of B or C to a non-zero value.

Set the unknown value to zero.

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NOTE: In the diagram of the parameters box above, A=60, B=0, and C=20. This means we are solving the same problem as example 1 above. We are given an angle of 60º, and the hypotenuse is 20, and we are calculating the length of the adjacent side.

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Click the 'Update' button to refresh the diagram and calculations.

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#### Step 3Â  Adjust the size of the diagram

If the triangle diagram is too big to display properly on your computer screen, briefly press the F10 key to reduce its size. To make the diagram bigger, hold down a Ctrl key while you press F10.

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