3.10.1 Sine Rule

3.10.1 Sine rule

The sine rule (also known as the law of sines) is a trigonometric formula used to find the unknown sides or angles of a triangle.

• It applies to any triangle, whether it is a right triangle or not.

• The sine rule states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is equal to a constant value, which is the same for all sides and angles of the triangle.

• Mathematically, this can be written as:

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

where \( a, b \) and \( c \) are the lengths of the sides of the triangle, and \( A, B \) and \( C \) are the angles opposite those sides, respectively.

• The sine rule can be used to solve a variety of problems involving triangles.

• For example, if you know two sides of a triangle and the angle opposite one of those sides, you can use the sine rule to find the measure of the other angles and the length of the remaining side.

Worked example:

Suppose we have a triangle with sides \( a = 5 \), \( b = 7 \), and angle \( A = 60 \) degrees.

We want to find the measure of angle \( B \) and the length of side \( c \).

Using the sine rule, we can write

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

Substituting in the given values, we get

\( \frac{5}{sin 60} = \frac{7}{sin B} = \frac{c}{sin C} \)

Simplifying, we get

\( \frac{5}{sin 60} = \frac{7}{sin B} \)

\( sinB = \frac{7\times sin60}{5} \)

\( sinB ≈ 0.866 \)

\( B≈60.6° \)