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How to find unknown opposite or adjacent sides in right triangles using tangent ratios.

 This topic is part of the TCS FREE high school mathematics 'How-to Library'. It shows you how to find unknown opposite or adjacent sides in right triangles using tangent 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 tangent ratio can be used to calculate the length of one of the shorter sides (legs) of a right triangle. To do this you need the tangent ratio for one of the smaller angles, and the length of one leg of the triangle.

There are two types of these problems, depending on whether you are finding the ‘opposite’ or ‘adjacent’ side. Two examples will explain the problem types:

Problem type 1: Finding the opposite side.

Example 1: Use the tangent ratio to find the unknown side in this triangle:

Solution:

But tan35º  = 0.700208  (From a table of tangent ratios, or scientific calculator with a ‘tan’ button.)

so x  =  10 × 0.700208

= 7.00208

Problem type 2: Finding the adjacent side.

Example 2: Use the tangent ratio to find the unknown side in this triangle:

Solution:

But tan40º  = 0.8391  (From a table of tangent ratios, or scientific calculator with a ‘tan’ button.)

Method:

Maths Helper Plus shows you to how to solve your own right triangles for unknown short sides (legs) by using tangent 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' = adjacent side to A.

edit box 'C' = opposite side to A.

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=35, B=10, and C=0. This means we are solving the same problem as example 1 above. We are given an angle of 35º, and the adjacent side to this angle of length 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.