WHAT IS THE SINE RULE – MINDSPARK

SINE RULE

Trigonometry is an important branch of mathematics. We study the different trigonometric ratios that help us in problems related to the sides and angles of triangles. The six trigonometric ratios are sin, cos, tan, cosec, sec and cot.

The sine rule says that the sides of any triangle are in proportion to the sin of the opposite angles. The formula is: 

\frac{a}{\operatorname{Sin} A}=\frac{b}{\operatorname{Sin} B}=\frac{c}{\operatorname{Sin} C}

Where a, b and c are sides and A, B and C are angles of a given triangle.

Before we go any further we should know about Pythagoras theorem which helps in solving questions related to right-angled triangles. The theorem is

\text { Perpendicular }^{2}+\text { Base }^{2}=\text { Hypotenuse }^{2}

Here, the side opposite to Ɵ is the perpendicular which is side a, side b is the base and side c is the hypotenuse.

We generally associate the trigonometric ratios with a right-angled triangle. Sin theta is the ratio of the side opposite the considered angle (also known as the perpendicular) and the hypotenuse. Cos is the complementary angle of sin. Cos is the ratio of the adjacent side of the considered triangle (also known as the base) and the hypotenuse. 

The sine rule can be manipulated to find sides as well as angles. It can be used not just in right- angled triangles but any other triangle where we know the side and its opposite angle.

Consider the following example: 

Given a triangle PQR with two angles as 30° and 60° and the length of one side as 7 cm.

Find the value of p.

Solution:

By Sine Rule we have, \frac{p}{\operatorname{Sin} P}=\frac{q}{\operatorname{Sin} Q}=\frac{r}{\operatorname{Sin} R}

Putting all the values we know,

\frac{p}{\operatorname{Sin} 30^{\circ}}=\frac{7}{\operatorname{Sin} 90^{\circ}}=\frac{r}{\operatorname{Sin} 60^{\circ}}

(Since the sum of all angles of a triangle is 180° so angle Q = 180° – (60° + 30°) = 90°, using algebra)

\frac{p}{1 / 2}=\frac{7}{\sqrt{3} / 2} \text {, which give } p=\frac{7}{\sqrt{3}}.

Note:

There is also a cos rule which is helpful in finding angles and sides as well. The formula is:

a^{2}=b^{2}+c^{2}-(2 b c \times \operatorname{Cos} A)

where a is the side we are trying to find and the sides b and c are known.