Law of Sines

What You’ll Learn

In this lesson you’ll learn the Law of Sines and how to use it to find missing sides and angles in non-right triangles.

The Concept

The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant:

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

This works for any triangle, not just right triangles. You need at least one side and its opposite angle, plus one other piece of information.

It’s especially useful when you have:

• Two angles and one side (AAS or ASA)
• Two sides and a non-included angle (SSA, the “ambiguous case”)

The ambiguous case (SSA) is worth knowing about: depending on the given values, there might be zero, one, or two valid triangles. This happens because the sine function gives the same value for an angle and its supplement (sin θ = sin(180° - θ)).

Worked Example

Example 1 (AAS): Find side b

In triangle ABC, angle A = 40°, angle B = 65°, and side a = 12 cm.

First, find angle C:

$$C = 180° - 40° - 65° = 75°$$

Now use the Law of Sines to find b:

$$\frac{12}{\sin 40°} = \frac{b}{\sin 65°}$$ $$b = \frac{12 \times \sin 65°}{\sin 40°} \approx \frac{12 \times 0.9063}{0.6428} \approx 16.92 \text{ cm}$$

Example 2 (SSA - Ambiguous Case)

Given angle A = 30°, side a = 10, side b = 15. Find angle B.

Using the Law of Sines:

$$\frac{10}{\sin 30°} = \frac{15}{\sin B}$$ $$\sin B = \frac{15 \times \sin 30°}{10} = \frac{15 \times 0.5}{10} = 0.75$$ $$B = \sin^{-1}(0.75) \approx 48.6°$$

But since sin(48.6°) = sin(131.4°), there’s a second possible value: B = 131.4°.

Check both: if B = 131.4°, then A + B = 30° + 131.4° = 161.4°, leaving C = 18.6°. That’s valid (all angles positive and sum to 180°). So this SSA case produces two valid triangles.

Real-World Application

The Law of Sines is used in:

• Surveying and land measurement (finding distances across rivers or inaccessible areas)
• Navigation and astronomy (calculating distances between celestial objects)
• Engineering and construction (designing structures with non-right angles)
• Forensics and accident reconstruction

Example: a surveyor measures two angles and one side across a river and uses the Law of Sines to calculate the river’s width without crossing it.

You’ve Got This

The Law of Sines is a powerful tool for any triangle. Remember: side / sin(opposite angle) is constant. When given SSA, be careful about the ambiguous case. Draw the triangle and check whether one or two solutions exist. This is one of the most practical tools in trigonometry.

Quiz

The Law of Sines states:

• A.$a^2 = b^2 + c^2 - 2bc\cos A$
• B.$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
• C.$\sin A + \sin B = \sin C$
• D.$a^2 + b^2 = c^2$

The Law of Sines is most useful when you know:

• A.Only the hypotenuse
• B.All three sides only
• C.Only right triangles
• D.Two angles and one side, or two sides and a non-included angle

In the ambiguous case (SSA), how many possible triangles can exist?

• A.0, 1, or 2
• B.Always 1
• C.Always 2
• D.Only 3

Why does the ambiguous case occur with SSA?

• A.Because tangent has period $\pi$
• B.Because cosine is always positive
• C.Because $\sin\theta = \sin(180° - \theta)$
• D.Because all triangles are right triangles

In triangle ABC with A = 50°, B = 70°, and a = 20, side b equals approximately:

• A.$\approx 18.3$
• B.$\approx 24.5$
• C.$\approx 30.1$
• D.$\approx 15.0$