Sine Rule in Trigonometry with Formula and Uses

It is easy to find the missing angle when the other two angles of a triangle are given, but the real confusion begins when you have to find a missing angle when given one angle and two sides or vice-versa. Don’t worry; this doesn’t mean it's impossible. Our mathematicians have got it covered! In the 11th century, Ibn Mu'adh al-Jayyani presented the sine rule, also known as the law of sines. 

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But before jumping on the law of sines, let's understand the meaning of the term sine.

Let’s take a right triangle $\text{XYZ}$ 

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Considering that $\text{XZ}$ is the right triangle's hypotenuse, the sine of the angle $\text{YZA}$ equals the$\frac{\text{Length XY}}{\text{Length XZ}}$ ratio.

Sine < $\text{YZA}$=$\frac{\text{XY}}{\text{ XZ}}$

Likewise, the sine of angle $\text{YXZ}$ equals the $\frac{\text{Length YZ}}{\text{Length XZ}}$ ratio.

Therefore, $\text{YXZ}$=  $\frac{\text{YZ}}{\text{XZ}}$

As a result, the sine of an angle is the ratio of the angle's opposing side length to the length of the hypotenuse.

What is Law of Sines?

Sine rule is one of the most useful mathematical laws, especially when it comes to solving triangles. The sine rule calculator helps to calculate the sides and angles of a triangle when not enough information is provided.

The rule of sines connects the side length ratios of triangles to their opposing angles. This proportion is true for all three sides and opposing angles. We can use the sine rule in triangle to get the missing angle or side of any triangle using the existing data.

Definition of The Law of Sines

The ratio of a triangle's side and matching angle equals the diameter of the triangle's circumcircle. As a result, the sine law may be written as, $\frac{\text{a}}{\sin A} =\frac{\text{b}}{\sin B}=\frac{\text{c}}{\sin C} $

where a,b and c are the lengths of the sides of a triangle and A, B and C are the opposite angles. 

To use the area of triangle sine rule, one must know either two angles and one side of the triangle or two sides and one angle. Along with this, two sides and an angle opposite one of them can be used to calculate the angle through the sine rule. 

How to Use the Law of sines?

If either one angle and the matching opposite side of a triangle are known, and either the side or angle of another side is also given, the Law of Sines may locate the missing information. The greatest approach to learn how is to practice using an example. Consider the triangle below. Using what we've learnt, we'll solve for side "c."

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Here the given items are: 

B= 35°

C= 105°

b = 7

• The first step is to identify what is given and note it down in a crisp form

• The second step is to determine what needs to be calculated; we have to find side “c”.

• The next step is to put all the given numbers into the sine equation, which is going to be: 

$\frac{\text{a}}{\sin A} =\frac{7}{\sin (35^{\circ})}=\frac{\text{c}}{\sin(105)}$ 

• By applying cross multiplication and replacing the values for sin B and sin C with numerical estimates, and simply find the angle and corresponding side by solving it. 

Explanation of The Sine Rule Proof

First, identify the triangle ABC in the normal manner so that angle A is the opposite side a.

Angle B is the opposite of side b, while angle C is the opposite of side c.

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Using basic trigonometry, 

$\sin(B) = \frac{\text{h}}{\text{c}}$    &  $\sin(c) = \frac{\text{h}}{\text{b}}$ 

Therefore, h= c sin (B) & h= b sin (c) respectively. 

So, c sin (B) = b sin (C)

Dividing both sides by bc, we get: $\frac{\sin (B)}{b} =\frac{\sin (C)}{c}$

Similarly, using an altitude from B or C, we would also include $\frac{\sin (A)}{a}$

Thus, the sine rule: $\frac{\sin (A)}{a}= \frac{\sin (B)}{b} =\frac{\sin (C)}{c}$

Which can be written as:  $\frac{a}{\sin (A)}= \frac{b}{\sin {B}} =\frac{c}{\sin {C}}$

Hence the law of sine proved!