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How to find an unknown opposite side or hypotenuse using sine ratios.

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

### Theory:

The sine 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 sine 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 opposite side.

Example 1: Use the sine ratio to find the unknown opposite side ‘x’ in this triangle:

Solution:

But sin30º  = 0.5  (From a table of sine ratios, or scientific calculator with a ‘sin’ button.)

so x  =  10 × 0.5

=  5

Problem type 2: Finding the hypotenuse.

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

Solution:

But sin 40º  = 0.642788  (From a table of sine ratios, or scientific calculator with a ‘sin’ button.)

### Method:

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

#### Step 2  Display the parameters box

Press the F5 key to display the parameters box:

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

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.

edit box 'B' = opposite side to A.

edit box 'C' = hypotenuse.

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

Set the unknown value to zero.

NOTE: In the diagram of the parameters box above, A=30, B=0, and C=10. This means we are solving the same problem as example 1 above. We are given an angle of 35º, and the hypotenuse is 10. We are calculating the length of the opposite side.

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

#### Step 3  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.