# Sine Formula

## Sine Formula

A particularly distinctive area of Mathematics called trigonometry examines connections between triangles’ sides and angles. In the study of geometry, trigonometry is often used. Additionally, there are some fascinating connections between trigonometry and other subjects of mathematics like calculus and logarithms. The crucial Sine Formula and cosine formula of trigonometry may be used to determine the answer for an oblique triangle. The important Sine Formula will be covered by students in the article that is posted on the Extramarks website and mobile application, along with its derivation and examples.

## What is Sine Formula?

Think about an oblique triangle, which lacks a correct angle. As a result, it is either a triangle with all acute angles or a triangle with one obtuse angle. It is especially helpful when filling in the blanks in a triangle.

For instance, using the Sine Formula, students may determine any or all of a triangle’s three angles if its three sides are known. The Sine Formula also enables us to determine the length of the third side when two sides and the angle between them are known.

## The Law of Sine

The following situations involve the use of the Sine Formula:

CASE-1: Triangle with two angles and one side, either AAS or ASA.

CASE-2: Triangle with two sides and an SSA that is not included.

According to the Sine Formula, a triangle’s sides are proportionate to the sines of its opposing angles. In mathematical form, the Sine Formula has been discussed extensively on the Extramarks website and mobile application.

The ratio of the triangle’s opposite side to the angle divided by the hypotenuse is known as the Sine Formula. The mnemonic SOH-CAH-TOA makes it simple to recall this ratio as well as the ratios for the other trigonometric functions. Those are,

• Sine Formula is opposite to the hypotenuse, or SOH.
• Cosine Adjacent Over the Hypotenuse (CAH)
• TOA stands for Tangent is Opposed to the Adjacent.

If students need to know how to measure an angle or solve an issue requiring height or distance, they can utilize this ratio.

### Derivation of the Sine Formula

Create an altitude via B and call it ashB to get the formula. The Sine Formula may be obtained by expressing hB in terms of the side and sine of the angle.

The Sine Formula specifies how many sides there are in a triangle and how their individual sine angles are equal. The sine law, sine rule, and Sine Formula are other names for the sine law.

The side or unknown angle of an oblique triangle is found using the law of sine. Any triangle that is not a right triangle is referred to as an “oblique triangle.” At least two angles and their corresponding side measurements should be used at once for the Sine Formula to function.

In order to determine the triangle’s unknown lengths or angles, we apply the Sine Formula. Sine Rule, Sine Law, or Sine Formula are other names for it. The following is a Sine Formula for the law of sines that may be used to determine an unknown triangle angle:

Sin A/a equals Sin B/b and Sin C/c.

The fraction is switched around in this instance. It implies a/sin A rather than taking a/sin A.

## Solved Examples for the Sine Formula

Question: Solve △PQR in which ∠ P = 63.5° and ∠Q = 51.2°  and r = 6.3 cm.

Solution: First, students need to calculate the third angle-

∠R = 180° – 63.5°- 51.2 °= 65.3°

Now, let’s calculate the sides:

6.3/Sin 65.3 = p/Sin 63.5

p = (6.3 × Sin 63.5)/Sin 65.3

p= 6.21 cm approximately.

Similarly, 6.3/Sin 65.3 = q/Sin 51.2

q = (6.3 × Sin 51.2) / Sin 65.3

q= 5.40 cm

∠R= 65.3°

p=6.21 cm

q=5.40 cm

Question: Given: △ABC, AB = c, BC = a and AC = b.

Answer: Construction: Draw a perpendicular, CD ⊥ AB. Then CD = h is the height of the triangle. “h” separates the △ ABC into two right-angled triangles, △CDA and △CDB.

To Show: a / b = Sin A / Sin B

Proof: In the  △CDA,

Sin A= h/b

And in  △CDB,

Sin B = h/a

Therefore, Sin A / Sin B = (h / b) / (h / a)= a/b

Thus, proved.

Likewise, students can prove, Sin B/ Sin C= b / c and so on for any pair of angles and their opposite sides.