Solution of Triangles: Sine Rule

The sine rule, also known as the law of sines, is a fundamental rule in trigonometry that relates the lengths of the sides of a triangle to the sines of its angles. This rule is particularly useful in solving for unknown parts of a triangle when some angles and sides are known. It applies to any triangle, whether it is acute, obtuse, or right-angled.

The Sine Rule Formula

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 the same for all three sides and angles in the triangle. The formula is expressed 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.

When to Use the Sine Rule

The sine rule is used in the following situations:

• When you know two angles and one side (AAS or ASA condition).
• When you know two sides and a non-included angle (SSA condition).

It is important to note that the SSA condition can sometimes lead to ambiguous cases where two different triangles can be formed (known as the ambiguous case of the sine rule).

Differences and Important Points

Here is a table summarizing the key points and differences when using the sine rule:

Aspect	Description
Applicability	The sine rule can be used for any triangle.
Known Quantities	Requires knowledge of at least one side and its opposite angle.
Formula	( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} )
Ambiguous Case	The SSA condition may lead to two possible solutions.
Use in Right-Angled Triangles	While applicable, it is often more convenient to use the Pythagorean theorem or basic trigonometric ratios.

Examples

Example 1: Solving for a Side

Given a triangle with sides ( a = 7 ) units, ( b = 5 ) units, and angle ( B = 30^\circ ), find the angle ( A ).

First, we use the sine rule to find the angle ( A ):

$$ \frac{a}{\sin A} = \frac{b}{\sin B} \implies \frac{7}{\sin A} = \frac{5}{\sin 30^\circ} $$

Since ( \sin 30^\circ = \frac{1}{2} ), we have:

$$ \frac{7}{\sin A} = \frac{5}{\frac{1}{2}} \implies \sin A = \frac{7}{10} $$

Now, we find the angle ( A ) by taking the inverse sine:

$$ A = \sin^{-1}\left(\frac{7}{10}\right) \approx 44.42^\circ $$

Example 2: Ambiguous Case

Consider a triangle with sides ( a = 8 ) units, ( b = 6 ) units, and angle ( A = 40^\circ ). We want to find angle ( B ).

Using the sine rule:

$$ \frac{a}{\sin A} = \frac{b}{\sin B} \implies \frac{8}{\sin 40^\circ} = \frac{6}{\sin B} $$

Solving for ( \sin B ), we get:

$$ \sin B = \frac{6 \cdot \sin 40^\circ}{8} $$

However, there are two possible values for ( B ) since ( \sin B ) could correspond to an acute or an obtuse angle. We need additional information to determine the correct angle ( B ).

In conclusion, the sine rule is a powerful tool in solving triangles, but it requires careful consideration of the given information and the potential for ambiguous cases. Understanding when and how to apply the sine rule is essential for solving problems in trigonometry and related fields.