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Sine Formula

Trigonometry is a very unique branch of mathematics that studies relationships between the sides and angles of triangles. Trigonometry is found all throughout the field of geometry. Also, trigonometry has some interesting relationships with other branches of mathematics like logarithms, and calculus. The solution for an oblique triangle can be found with the important sine formula and cosine formula of the trigonometry. In this article, we will discuss the important Sine formula with derivation and examples. Let us begin the topic!

What is Sine Formula?

Let us take an oblique triangle, i.e. a triangle with no right angle. Therefore, it is a triangle whose angles are all acute or a triangle with one obtuse angle. It is most useful for solving for missing information in a given triangle.

For example, if all three sides of the triangle are known, then Sine formula will allow us to find any or all of its three angles. Similarly, if two sides and the angle between these two sides is known, then the Sine formula allows us to find the third side length.

The Law of Sine

The Sine Rule is used in the following cases as follows:

  • CASE-1: Given two angles and one side in triangle i.e. AAS or ASA.
  • CASE-2: Given two sides and a non-included angle in triangle i.e. SSA.

The Sine Rule states that the sides of a triangle are in the proportional of the sines of the opposite angles. In form of mathematics:

\(\frac{a}{\sin A}= \frac{b}{\sin B} =\frac{c}{\sin C} \)


Derivation of the Sine Formula

To derive the formula, erect an altitude through B and termed it as\( h_B\). Expressing \(h_B\) in terms of the side and the sine of the angle will give the sine law formula.

\(\sin A = \frac{h_B}{c}\)

\(h_B = c \sin A \)

\(\sin C = \frac{h_B}{a}\)

\(h_B = a \sin C \)

Equate the two \(h_B’s \)above:

\(h_B = h_B\)

c \(\sin A = a \sin C\)

\(\frac{c}{\sin C} = \frac{a}{\sin A}\)

To include angle B and side b in the above relationship, then construct an altitude through C and termed it as \(h_C\).

\(\sin A = \frac{h_C}{b}\)

\(h_C = b \sin A\)

\(\sin B = \frac{h_C}{a}\)

\(h_C = a \sin B\)

\(h_C = h_C\)

\(b \sin A = a \sin B\)

\(\frac{b}{\sin B} = \frac{a}{\sin A}\)

\(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)

Therefore, the sine function is the ratio of the side of the triangle opposite to angle and divided by the hypotenuse. Easy way to remember this ratio along with the ratios for the other trigonometric functions is with the mnemonic as SOH-CAH-TOA. Which are,

  • SOH = Sine is Opposite over the Hypotenuse
  • CAH = Cosine is Adjacent over the Hypotenuse
  • TOA = Tangent is Opposite over the Adjacent

This ratio can be used to solve problems involving distance or height, or if you need to know an angle measure.

Solved Examples for the Sine Formula

Q.1. Solve triangle PQR in which \(\angle P = 63.5^{\circ} and \angle Q = 51.2^{\circ}\)  and r = 6.3 cm.

Solution: First, calculate the third angle.

\(\angle R= 180 – (63.5 + 51.2) = 65.3^{\circ}\)

Next, calculate the sides.

\(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)

\(\frac{6.3}{\sin 63.5} = \frac{p}{\sin 63.5}\)

p = 6.21 cm

Similarly q = 5.4 cm

Thus \(\angle R = 65.3^{\circ}\)

p = 6.21 cm

q = 5.4 cm

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