Law of Sines Explained

The Law of Sines

Match Every Side with Its Opposite Angle

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

Side \(a\) is opposite angle \(A\), side \(b\) is opposite angle \(B\), and side \(c\) is opposite angle \(C\).

The Most Important Setup

Keep the Side-Angle Pairs Together

Each lowercase side is paired with the uppercase angle directly across from it.

\(A\) ↔ \(a\)

Angle \(A\) is opposite side \(a\).

\(B\) ↔ \(b\)

Angle \(B\) is opposite side \(b\).

\(C\) ↔ \(c\)

Angle \(C\) is opposite side \(c\).

Think Across the Triangle

Do not match a side with an angle that touches it. The correct angle is always directly across from the side.

Visual Model

See the Opposite Pairs

In triangle \(ABC\), each side is named with the lowercase letter of the angle directly across from it.

Uppercase and Lowercase Matter

Uppercase letters \(A\), \(B\), and \(C\) name angles. Lowercase letters \(a\), \(b\), and \(c\) name the sides opposite those angles.

Choosing the Method

When Should You Use the Law of Sines?

The Law of Sines is especially useful when you know at least one complete side-angle pair.

ASA

Angle-Side-Angle

Two angles and the side between them are known.

\[ A,\ c,\ B \]

Find the third angle first, then use the Law of Sines.

AAS

Angle-Angle-Side

Two angles and a non-included side are known.

\[ A,\ B,\ a \]

Find the third angle if needed, then build a side-angle proportion.

SSA

Side-Side-Angle

Two sides and an angle that is not between them are known.

\[ a,\ b,\ A \]

This is the ambiguous case, so there may be zero, one, or two possible triangles.

Method Check

Law of Sines or Law of Cosines?

1

Look for a complete pair

If you know a side and its opposite angle, the Law of Sines is often the best choice.

2

ASA or AAS?

Use the Law of Sines after finding the third angle if necessary.

3

SSA?

The Law of Sines can work, but check for the ambiguous case.

4

SAS or SSS?

Those usually point to the Law of Cosines instead.

Worked Examples

Solving with the Law of Sines

The key is to build a proportion using two matching side-angle pairs.

Example 1

Find a Missing Side

AAS

In triangle \(ABC\), \(A=42^\circ\), \(B=71^\circ\), and \(a=10\). Find \(b\).

1. Match the known pair.

   Side \(a=10\) is opposite angle \(A=42^\circ\).

2. Match the unknown pair.

   Side \(b\) is opposite angle \(B=71^\circ\).

3. Write the proportion.
   \[ \frac{10} {\sin(42^\circ)} = \frac{b} {\sin(71^\circ)} \]
4. Solve for \(b\).
   \[ b = \frac{ 10\sin(71^\circ) }{ \sin(42^\circ) } \]
5. Evaluate.
   \[ b \approx 14.13 \]

Answer \[ \boxed{b\approx14.13} \]

Example 2

Find a Missing Angle

SSA

In triangle \(ABC\), \(A=36^\circ\), \(a=8\), and \(b=11\). Find angle \(B\).

1. Use the known pair.

   Side \(a=8\) is opposite angle \(A=36^\circ\).

2. Match side \(b\) with angle \(B\).
   \[ \frac{8} {\sin(36^\circ)} = \frac{11} {\sin B} \]
3. Solve for \(\sin B\).
   \[ \sin B = \frac{ 11\sin(36^\circ) }{ 8 } \]
4. Use inverse sine.
   \[ B = \sin^{-1} \left( \frac{ 11\sin(36^\circ) }{ 8 } \right) \]
5. Evaluate.
   \[ B \approx 53.9^\circ \]

First Possible Angle \[ \boxed{ B\approx53.9^\circ } \]

The SSA Ambiguous Case

Sometimes One Answer Is Not Enough

When you are given two sides and a non-included angle, the Law of Sines may produce two different triangles.

Why Does This Happen?

Sine has the same positive value for an acute angle and its supplementary obtuse angle.

\[ \sin(\theta) = \sin(180^\circ-\theta) \]

So if inverse sine gives you one angle, you should check its supplement whenever you are solving an SSA triangle.

Continue Example 2

Check for a Second Triangle

SSA

We already found the first possible value of angle \(B\):

\[ B_1 \approx 53.9^\circ \]

1. Find the supplementary angle.
   \[ B_2 = 180^\circ - 53.9^\circ \] \[ B_2 \approx 126.1^\circ \]
2. Check the angle sum.

   The given angle is \(A=36^\circ\). Test the possible second angle:

   \[ 36^\circ + 126.1^\circ = 162.1^\circ \]
3. Decide whether the triangle is possible.

   Since \(162.1^\circ<180^\circ\), there is still room for a positive third angle.

   \[ C = 180^\circ - 162.1^\circ \] \[ C \approx 17.9^\circ \]

Triangle 1 \[ B_1 \approx 53.9^\circ \]

Triangle 2 \[ B_2 \approx 126.1^\circ \]

Two Triangles Are Possible

Both possible values of \(B\) produce a valid triangle, so this SSA problem has two solutions.

SSA Outcomes

An SSA Problem Can Have 0, 1, or 2 Triangles

0

No Triangle

The given measurements cannot form a valid triangle.

\[ \sin B>1 \]

A sine value greater than 1 is impossible.

1

One Triangle

Only one possible angle produces a valid triangle.

\[ A+B<180^\circ \]

The supplementary angle fails the triangle angle-sum test.

2

Two Triangles

Both the inverse-sine angle and its supplement are valid.

\[ B_2 = 180^\circ-B_1 \]

Both possibilities leave room for a positive third angle.

SSA Checklist

When You Use Inverse Sine, Check the Supplement

1

Use the Law of Sines to find the first possible angle.

2

Find its supplement using \(180^\circ-\theta\).

3

Add the supplementary angle to the angle you were originally given.

4

If the sum is less than \(180^\circ\), a second triangle may exist.

Remember SSA is the case that requires the extra ambiguous-case check.

Check Your Reasoning

Common Law of Sines Mistakes

Most errors come from pairing the wrong side and angle or forgetting to check the ambiguous case.

Pairing the Wrong Side and Angle

Side \(a\) must be paired with angle \(A\), side \(b\) with angle \(B\), and side \(c\) with angle \(C\).

Flipping Only One Ratio

You may write side over sine or sine over side, but every ratio in the equation must use the same orientation.

Forgetting Inverse Sine

If the unknown is an angle, isolate its sine value first, then use \(\sin^{-1}\).

Ignoring the SSA Ambiguous Case

When solving an SSA problem, always check whether the supplementary angle creates a second valid triangle.

Quick Review

Law of Sines Step-by-Step

1

Label the triangle

Match each side with the angle directly across from it.

2

Find a complete pair

Look for a known side and its known opposite angle.

3

Write two ratios

Use the known pair and the pair containing the unknown.

4

Solve

Cross multiply for a side or use inverse sine for an angle.

5

Check SSA

If the problem is SSA, test the supplementary angle.

Core Formula:

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

Ready to Practice?

Try Law of Sines Problems

Practice matching opposite pairs, finding missing sides, finding missing angles, and checking the SSA ambiguous case.

Practice Law of Sines Review SOH-CAH-TOA

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