Law Of Sine

The laws of sine also known as the rule of sine, is used to solve triangles

a sin A = b sin B = c sin C

law of sine

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a, b, and c are sides of the triangle and A, B and C are the angles of the triangle. The ‘a’ side which is facing the angle A, the side b is facing angle B and the c side is facing the angle C.

It is clear that whenever we divide the ‘a’ side, by sine, of the angle A which is equal to the b side, which is divided by the sine angle B and also the same as the c side which is divided by the sine angle C.

Let us calculate now as the triangle has angle A of 62.2 degrees, angle B of 33.5 degrees and angle C of 84.3 degrees with three sides of a, b, and c which measures 8, 5 and 9 respectively.

Solve the aSin A = 8sin(62.2) = 9.04

bsin B = 5sin(33.5) = 9.07
csin C = 9 sin(84.3) = 9.03

Thus, these all answers are the same. Now you can also see that

a sin A = b sin B = c sin C

This law of sines is used to find out the following

  • AAS or ASA: One side and two angles are known. The other side must be solved.
  • SSA: One angle and two sides are known. The other angle must be solved.

Example 1:

Find out the side c.

example 1-2

According to the law of sine,

a/sin A = b/sin B = c/sin C

Then input the values in the formula given : a/sin A = b/sin 35 = c/sin 105

Now just avoid r/sin A, which is not useful to us – 7/sin 35 = c/sin 105

Then we can use our skills of algebra for rearranging and solving it

Swap the sides = c/sin 105 = 7/sin 35

Multiply sin 105 in both of the sides which will then be seen as

c = 7/sin 35 * sin 105

Now by this, we will calculate c.

c = 11.8

Example 2:

Find out the angle B.

example 2-3

Start with: sin A / a = sin B / b = sin C / c

Put in the values we know: sin A / a = sin B / 4.7 = sin(63°) / 5.5

Ignore “sin A / a”: sin B / 4.7 = sin(63°) / 5.5

Multiply both sides by 4.7:sin B = (sin(63°)/5.5) × 4.7

Calculate: sin B = 0.7614…

Inverse Sine: B = sin−1(0.7614…)

B = 49.6°