SIN 90° – FORMULA, DERIVATION AND EXAMPLES

SIN 90°

Trigonometry is an important branch of mathematics that helps us deal with the sides and angles of a triangle. The trigonometric ratios are the ratios of two sides and Sine is a trigonometric ratio which is simply the ratio of the side opposite to the angle divided by the hypotenuse. 

In the above right-angled triangle, the sides are denoted by a, b and c where a is the perpendicular, b is the base and c is the hypotenuse.

Now, the trigonometric ratio sin is written as the ratio of the side opposite the angle to the hypotenuse and can be written as

Sin Ɵ = Side opposite to the angle / Hypotenuse

Here, the 90° angle between sides a and b is opposite the hypotenuse. So when we put the formula, we get

Sin 90° = c / c = 1

There are six trigonometric ratios. They are Sine(sin), Cosec(cos), Secant(sec), Tangent(tan) and Cotangent(cot). The table below will show how to find the values of the ratios. 

The common angles for which the values are standardly known are 0°, 30°, 45°, 60° and 90°. The term π which is simply interpreted as the 180° can be used to write these standard angles. The table below shows the value of the six trigonometric ratios for the standard angles:

As we can see, the table also shows the value of Sin 90° = 1.

DERIVING THE VALUE OF SIN 90°

We will use a unit circle to derive the value of sin 90°.

Draw a circle with centre O and radius 1 unit. 

Now, make a line on the x-axis and then another line that forms the angle Ɵ and name that Q. Draw a line perpendicular to the x-axis from Q. In order to derive the value of sin 90°, we will need to make the angle Ɵ as 90°. We will start by shifting OQ and measuring the angles by shifting anti-clockwise on the circle towards the y-axis. At a point, the OQ line will coincide with the y-axis which will make the value of side opposite Ɵ as 1 and the angle as 90°.

Thus, giving us the value of sin 90° = 1.

Let us look at an example.

For example: What is the value of hypotenuse if sin 90° is 1 and the length of the opposite side is 7 units?

Ans: Sin 90° = length of the opposite side/ hypotenuse

1 = 7 / Hypotenuse

Therefore, 7 = hypotenuse. 

Understand that, the opposite side to sin 90° is the hypotenuse only so through these calculations we get the same answer for the hypotenuse.