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Introduction to the cosine ratio.

 This topic is part of the TCS FREE high school mathematics 'How-to Library'. It introduces the 'cosine ratio' for right angled triangles. (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:

Consider this right angled triangle:

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Side ‘p’ is adjacent to angle ‘Q’, and side ‘q’ is adjacent to angle ‘P’.Â  (‘Adjacent’ means ‘beside’).

In right angled triangles, the term 'adjacent' is never used to refer to the longest side opposite to the right angle. This is always called the ‘hypotenuse’. Only the shorter sides, the ‘legs’ of the right triangle, can be 'adjacent' sides.

For an angle less than 90º in a right angled triangle, the following ratio:Â

 Â is called the ‘cosine ’ of the angle.Â

For example, the cosine ratio of angle ‘P’ above can be found as follows:Â Â Â  Â

Consider the triangle below:

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The hypotenuse in a right triangle is always larger than the adjacent side, so for angles greater than zero but less than 90º the cosine ratio will be less than 1. For angles outside of these limits, the sine ratio can have values from -1 to 1.

### Â Method:

Maths Helper Plus can calculate cosine ratios if the adjacent side and hypotenuse lengths of a right triangle are given. It will show the calculations and display a labelled diagram.Â The diagram is drawn in the same orientation as the triangle in the example above, with angles labelled 'P' and 'Q'.

#### 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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edit box 'A' = the adjacent side

edit box 'B' = the hypotenuseÂ

edit box 'C' = the scale factor for the triangle side lengths. ( Leave set to 1)

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If edit box 'X' = 1, then the cosine ratio is calculated for angle ‘P’, the angle at the lower left of the diagram, and the adjacent side A is horizontal.

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If edit box 'X' = 2, then the cosine ratio is calculated for angle ‘Q’, the angle at the top of the diagram, and the adjacent side A is vertical.

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

To display the calculations and diagram for the examples in the 'Theory' section above, enter these settings:

Example (a), calculating sine 'P':Â  A = 12, B = 15, C = 1, X = 1

Example (b), calculating sine 'Q', A = 9, B = 15, C = 1, X = 2

#### 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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