Special Triangles

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In Geometry, there are two special triangles. The special triangles are the triangles in which the angles are either 30º-60º-90º or 45º-45º-90º. The triangles forming these angles are called special triangles because when their values are calculated using the six trigonometric functions, then exact values are produced instead of decimal numbers. Therefore using any of these angles and the length of any one side of the triangle, the lengths of other sides can be easily calculated using basic trigonometric functions such as sine, cosine or tan of the special angles.

Example 1: The side AC is the hypotenuse in the right triangle ABC. If the hypotenuse of this triangle is 4cm, then what is the measure of side BC if the measure of angle C is 60º?

Here triangle ABC is a special triangle because one of its angle measures 60º which implies that the angles

of this triangle are in the form of 30º-60º-90º.

The trigonometric function, cos(C) = (adjacent side)/ (hypotenuse) = BC/AC

This gives: cos(60)= BC/ 4==> 1/2= BC/ 4==> BC= 4 * 1/2

Therefore the measure of the side, BC= 2cm


Example 2: The side PR is the hypotenuse in the right triangle PQR. If the measure of side PQ is 9cm, then what is the measure of the side QR if measure of angle R is 45º?


Here triangle PQR is a special triangle because one of its angle measures 45º which implies that the angles of this triangle are in the form of 45º-45º-90º.

The trigonometric function, tan(R) = (opposite side)/ (adjacent side) = PQ/QR

This gives: tan(45)= 9/ QR==> 1= 9/ QR==> QR= 1* 9

Therefore the measure of the side, QR= 9cm



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