SIN 30° – VALUE AND DERIVATION

SIN 30°

Sine (sin) is a trigonometric ratio. What are trigonometric ratios? Trigonometry is a branch of mathematics that deals with the sides and angles of triangles. So trigonometric ratios are the ratios of the sides of a right angles triangle. Now, a right-angled triangle looks like this:

The side opposite to the 90 degree angle is called the hypotenuse in the right-angled triangle. 

Here in this triangle,

a – perpendicular (the side opposite to Ɵ)

b – base

c – hypotenuse

The trigonometric ratio sine (sin) is simply the ratio of the side opposite the angle to the hypotenuse and can be written as:

Sin Ɵ = Perpendicular / Hypotenuse

So here Sin Ɵ = a / c

There are 6 trigonometric ratios namely sin, cos, tan, cosec, sec and cot. See the table below to find the values of these ratios and also their reciprocals.

As can be seen, cosec is simply the reciprocal of sin as the formula to find cosec is the reciprocal of sin. 

Though these ratios can be calculated for all angles, there are some standard examples such as 0°, 30°, 45°, 60° and 90°. These ratios can also be written in terms of π which here is interpreted as 180° angle. The table below shows the value of these standard angles:

The table shows that the value of sin 30° is ½ or 0.5.

DERIVATION OF SIN 30°

There are two methods of deriving sin 30°. Let us see both the methods below:

Method 1: Let’s take an equilateral triangle. An equilateral triangle has all the angles equal to 60° and all sides are also equal. 

Step 1. Draw a line perpendicular to the side BC. Also, since it is an equilateral triangle, let the length of the sides be ‘2a’ so AB = BC = AC = 2a.

Step 2. The perpendicular divides the side BC into half thus making BD= DC = a and also bisects the angle at A making it 30°.

The triangle ABD is a right-angled triangle at D.

Sin 30° = Side opposite to the angle 30°/ Hypotenuse 

Sin 30° = BD/ AB

Sin 30° = a/2a = ½ or 0.5

Hence, the value of Sin 30°= ½.

Method 2: The value can be derived by construction as well. 

Step 1. Draw a straight line from a point P. Use a protractor and ruler to draw a 30° angle from point P. 

Step 2. Since we need to know the lengths so use the compass and set the length as 6 cm. This point will come out to be Q.

Step 3. Draw a line perpendicular to the original horizontal line and mark the point R on where it intersects the original line. When you use a ruler to measure the length of QR, you will find the value as 3 cm. 

So, 

Sin 30° = Perpendicular or side opposite the angle / Hypotenuse

Sin 30° = 3 / 6

Sin 30° = ½

THE LAW OF SINE

The law of sine (sin) becomes helpful when we solve triangles.  The law states that the sides of any triangle are in proportion to the sine of the opposite angles. The law can be written as:

a/ Sin A = b/ Sin B = c/ Sin C

Where a, b and c are sides with angles A, B and C opposite to each other.  

This law is useful in two scenarios. Firstly, when we know two angles and one side and secondly, two sides and a non-included angle which is an angle that doesn’t lie between the sides. 

For example: Given a triangle PQR with two angles as 30° and 60° and the length of one side as 7 cm. 

We have to find the value of p.

Putting all the values we know, 

p/ Sin P = q/ Sin Q = r/ Sin R

p/ Sin 30° = 7/ Sin 90° = r/ Sin 60°

p/(1/2) = 7/ 1 = r/ √3

(Since the sum of all angles of a triangle is 180° so angle Q = 180° – (60° + 30°) = 90°

Using algebra,  we get

 p = 7/2