Law of Sines Triangle – Definition, Formula, Proof & Problems

# Law of Sines Triangle – Definition, Formula, Proof & Problems

## Law of Sine

The law of sine states that the sine of an angle is equal to the ratio of the length of the opposite side to the length of the hypotenuse.

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## Law of Sine Definition

The law of sine states that for any given angle, the length of the sine function’s curve is proportional to the length of the opposite side. In mathematical terms, this law states that the sine of an angle is equal to the length of the opposite side divided by the length of the hypotenuse.

## Law of Sines Formula

The law of sines states that the ratio of the length of the sine of an angle to the length of the opposite side is equal to the ratio of the lengths of the other two sides.

## Law of Sine Proof

A proof of the law of sine can be found using the Pythagorean theorem.

In the diagram, A is the angle of the triangle and b is the length of the side opposite A. The length of the other two sides is c and a, respectively.

The Pythagorean theorem states that:

a2 + b2 = c2

In this triangle, a is the length of the hypotenuse and b and c are the lengths of the two other sides. Substituting these values into the Pythagorean theorem, we get:

c2 = a2 + b2

c2 – a2 = b2

Now, using the law of sine, we can rewrite this equation as:

sin(A) = b/c

## Law of Sine – Ambiguous Case

In the ambiguous case, the law of sine states that the sine of an angle is the same as the sine of its complementary angle.

## When to Use the Law of Sines?

The law of sines can be used to solve problems in geometry, including problems involving triangles.

## Law of sine problem

A boat travelling due east at 10 mph passes a lighthouse at an angle of 15° north of east. Find the speed of the boat when it is nearest to the lighthouse.

The boat is travelling at 11.6 mph when it is nearest to the lighthouse.

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