# TRIGONOMETRIC RATIOS OF SOME SPECIFIC ANGLES

## About "Trigonometric ratios of some specific angles"

Trigonometric ratios of some specific angles :

For certain specific angles such as 30°, 45° an d 60°, which are frequently seen in applications, we can use geometry to determine the trigonometric ratios.

## Trigonometric ratios of 30° and 60°

Let ABC be an equilateral triangle whose sides have length a (see the figure given below). Draw AD perpendicular to BC, then D bisects the side BC.

So, BD  =  DC  =  a/2 and   < BAD  =  <DAC  =  30°. Now, in right triangle ADB, <BAD  =  30° and BD  =  a/2.

In right triangle ADB, by Pythagorean theorem,

Hence, we can find the trigonometric ratios of angle 30° from the right triangle ADB. In right triangle ADB, <ABD  =  60°. So, we can determine the trigonometric ratios of angle 60°. ## Trigonometric ratio of 45°

If an acute angle of a right triangle is 45°, then the other acute angle is also 45°.

Thus the triangle is isosceles. Let us consider the triangle ABC with <B  =  90°, <A  =  <C  =  45° Then AB  =  BC. Let AB  =  BC  =  a.

By Pythagorean theorem,

AC²  =  AB² + BC²

AC²  =  a² + a²

AC²  =  2a²

AC  =  √2 x a

Hence, we can find the trigonometric ratios of angle 45° from the right triangle ABC. ## Trigonometric ratios of 0° and 90°

Consider the figure given below which shows a circle of radius 1 unit centered at the origin.

Let P be a point on the circle in the first quadrant with coordinates (x, y). We drop a perpendicular PQ from P to the x-axis in order to form the right triangle OPQ.

Let <POQ  =  θ, then

sin θ  =  PQ / OP  =  y/1  =  y  (y coordinate of P)

cos θ  =  OQ / OP  =  x/1  =  x  (x coordinate of P)

tan θ  =  PQ / OQ  =  y/x

If OP coincides with OA, then angle θ  =  0°.

Since, the coordinates of A are (1, 0), we have If OP coincides with OB, then angle θ  =  90°.

Since, the coordinates of B are (0, 1), we have The six trigonometric ratios of angles 0°, 30°, 45°, 60° and 90° are provided in the following table. After having gone through the stuff given above, we hope that the students would have understood "Trigonometric ratios of some specific angles"