Sine Theorem

In a triangle we can explore a number of relationships. These relation ships help us to solve a triangle. That is by knowing minimal quantities out of all the three angles and all the three sides, the remaining quantities can be figured out.

 When two sides and one angle (not the included angle) or two angles and any side are known the remaining parameters can be found. The sine theorem of triangle helps us in that.

Let us see what a sine theorem is.

The Statement of Sine Theorem-



Let ABC be any scalene type of triangle.

As per sine theorem, the following ratios are equal in any triangle.

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

This ratio is equal to the diameter of the circumscribing circle of the triangle.

Let us illustrate with an example.

The measures of two angles of a triangle are 30 degrees and 45 degrees. The measure of one side is 30 cm. Solve the triangle.

Let us assume A = 30o B = 45o and c = 30cm.

Using the property of sum of angles of a triangle,

C = 180o – (30o + 45o) = 105o

As per the sine theorem,

$\frac{a}{sin 30}$ = $\frac{b}{sin 45}$ = $\frac{30}{sin 105}$

Therefore,

a = 30 $\frac{sin 30}{sin 105}$ = 30 (0.52) = 15.6 cm (approximately)

b = 30 $\frac{sin 45}{sin 105}$ = 30 (0.74) = 22.2 cm (approximately)

Sine Theorem- Derivation



In the same diagram shown earlier, draw a perpendicular CD on AB and a perpendicular BE on AC.

In triangle ACD, CD = ACsin A = bsin A

In triangle BCD, CD = BCsinB = asin B

Therefore, bsinA = asinB

or,    $\frac{a}{sin A}$ = $\frac{b}{sin B}$

Similarly in triangle ABE, BE = ABsin A = csin A

and in triangle BCE, CE = BCsinC= asin C

Therefore, csinA = asinC

or,    $\frac{a}{sin A}$ = $\frac{c}{sin C}$

Combining both the results,

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