SIN 45° – VALUE AND DERIVATION

TRIGONOMETRY AND THE RATIO SIN 45°

Trigonometry (a branch of mathematics) helps us to understand the relationship between lengths of sides and angles of triangles. The ratio between two sides of a triangle is called the trigonometric ratios. There are six such trigonometric ratios. They are sin, cos, tan, cosec, sec and cot. We use the right angled triangle to understand these ratios. 

We use sin Ɵ (where Ɵ can be any value, here we will know about sin 45°) to denote the angle and the ratio is given as:

Sin Ɵ = side opposite the angle Ɵ/ hypotenuse.

Hypotenuse is simply the longest side of the right angled triangle and is opposite the 90° angle. 

Here, a is the side opposite the angle Ɵ, b is the base and c is the hypotenuse with the angle between side a and b being the 90° angle. The table below shows the value of all the six trigonometric ratios at different angles:

VALUE OF SIN IN THE 4 QUADRANTS

There are 4 quadrants that start from 0° and end at 360°. The first quadrant ranges from 0° to 90°, the second quadrant ranges from 90° to 180°, the third quadrant ranges from 180° to 270° and the fourth quadrant ranges from 270° to 360°.

The value of sin is positive in the first and second quadrant but negative in the third and fourth quadrant.

DERIVING THE VALUE OF SIN 45°

The value sin 45° can be easily derived.

In the right angled triangle PQR, angle Q is 90° and angle R is 45°. 

Since the sum of the angles of a triangle is 180° so the angle P is 45° (180° – 90° – 45° = 45°).

Now, 

PQ = QR = a

Using Pythagoras Theorem, 

(PQ)² + (QR)² = (PR)²

a² + a² = (PR)²

2a² = (PR)

√2a = PR

Sin 45° = opposite side / hypotenuse

Sin 45° = a /√2a

Sin 45° = 1/ √2

Hence, we have derived the value for sin 45°

TRIGONOMETRIC IDENTITIES

Apart from trigonometric ratios, we have trigonometric identities. These identities are standard just like the Pythagoras theorem. It helps in finding missing values if others are known.  The identities related to the trigonometric ratio sin are :

• Sin (A + B) = sin A cos B + cos A sin B

• Sin (A – B) = sin A cos B – cos A sin B

• Cos (A + B) = cos A cos B – sin A sin B

• Cos (A – B) = cos A cos B + sin A sin B