Laws of Sines
& Cosines

Master the two most powerful tools for solving any triangle: the Law of Sines and the Law of Cosines.

Geometry

Why Do We Need New Tools?

Basic trigonometric ratios ($\sin$, $\cos$, $\tan$) only work on right triangles. But most triangles in real life are not right triangles. The Laws of Sines and Cosines let us solve any triangle: acute, obtuse, or right.

Standard Triangle Labeling

What Can You Solve?

Find Missing Sides

Given enough angles and sides, calculate an unknown side length.

Find Missing Angles

Determine an unknown angle from known sides and angles.

Real-World Problems

Navigation, surveying, engineering, and physics all use these laws.

The Law of Sines

In any triangle $\triangle ABC$:

This says the ratio of each side to the sine of its opposite angle is the same constant for all three pairs. It works in every triangle: acute, right, and obtuse.

Proving the Law of Sines

The Law of Cosines

In any triangle $\triangle ABC$:

This is a generalization of the Pythagorean theorem. When $C = 90^\circ$, $\cos 90^\circ = 0$ and we get $c^2 = a^2 + b^2$. The term $-2ab\cos C$ is the "correction" for non-right triangles.

Three Forms of the Law of Cosines

Solving for side $a$

$$a^2 = b^2 + c^2 - 2bc\cos A$$

Solving for side $b$

$$b^2 = a^2 + c^2 - 2ac\cos B$$

Solving for side $c$

$$c^2 = a^2 + b^2 - 2ab\cos C$$

When to Use Which Law?

Given Information	Law of Sines	Law of Cosines
AAS (two angles + non-included side)	✓ Use this	Not needed
ASA (two angles + included side)	✓ Use this	Not needed
SSA (two sides + non-included angle)	✓ Ambiguous case	Alternative
SAS (two sides + included angle)	Not enough info	✓ Use this
SSS (three sides)	Not enough info	✓ Use this

Step-by-Step Problem Solving

1

Sketch and label. Draw the triangle and mark all known sides and angles.

2

Classify your knowns. Identify the case: AAS, ASA, SSA, SAS, or SSS.

3

Choose the right law. Law of Sines for AAS/ASA/SSA; Law of Cosines for SAS/SSS.

4

Substitute and solve. Plug values into the formula and solve for the unknown.

5

Check your answer. Verify all angles sum to $180^\circ$ and use the triangle inequality.

Example 1: Law of Sines (AAS)

Example 2: Law of Cosines (SAS)

Example 3: Finding an Angle (SSS)

The Ambiguous Case (SSA)

When you know two sides and a non-included angle (SSA), the Law of Sines can produce zero, one, or two valid triangles. This is because $\sin$ returns the same value for an angle and its supplement.

No Triangle

If $\sin B > 1$, the triangle is impossible.

One Triangle

If the supplement $180^\circ - B$ would make angles sum exceed $180^\circ$.

Two Triangles

Both $B$ and $180^\circ - B$ produce valid triangles.

Common Mistakes

✕ Wrong Law Choice

Using Law of Sines when you have SAS or SSS. You need an angle-side opposite pair to use Law of Sines.

✕ Forgetting the Ambiguous Case

With SSA, always check if $180^\circ - B$ also gives a valid triangle. Missing the second solution is a common error.

✕ Calculator in Wrong Mode

Using radians instead of degrees (or vice versa). Always check your calculator mode before computing trig values.

Summary Comparison

Feature	Law of Sines	Law of Cosines
Formula	$\frac{a}{\sin A} = \frac{b}{\sin B}$	$c^2 = a^2 + b^2 - 2ab\cos C$
Best for	AAS, ASA, SSA	SAS, SSS
Requires	An angle-side opposite pair	Two sides + included angle, or all sides
Ambiguous case?	Yes (SSA)	No
Special case	Relates to circle circumradius	Reduces to Pythagorean theorem

Question 1 of 5

In $\triangle ABC$, you know $A = 30^\circ$, $B = 80^\circ$, and $a = 12$. Which law should you use to find side $b$?

You have two angles and a non-included side (AAS). The Law of Sines is the correct choice because you have a complete angle-side opposite pair: $\frac{a}{\sin A} = \frac{b}{\sin B}$.

Question 2 of 5

In $\triangle PQR$, $p = 5$, $q = 8$, and $R = 60^\circ$. What is $r$?

This is SAS, so use Law of Cosines: $r^2 = 5^2 + 8^2 - 2(5)(8)\cos 60^\circ = 25 + 64 - 80(0.5) = 89 - 40 = 49$. Therefore $r = \sqrt{49} = 7$.

Question 3 of 5

In $\triangle ABC$, $a = 10$, $b = 10$, and $c = 10$. What is angle $A$?

Using Law of Cosines: $\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{10^2 + 10^2 - 10^2}{2(10)(10)} = \frac{100}{200} = 0.5$. So $A = \cos^{-1}(0.5) = 60^\circ$. An equilateral triangle has all angles equal to $60^\circ$.

Question 4 of 5

When solving an SSA problem with the Law of Sines, you find $\sin B = 0.85$. How many triangles are possible?

Since $0 < \sin B = 0.85 < 1$, angle $B$ could be $\sin^{-1}(0.85) \approx 58.2^\circ$ or $180^\circ - 58.2^\circ = 121.8^\circ$. You must check if both values produce a valid triangle (all angles sum to $180^\circ$). This is the ambiguous case.

Question 5 of 5

The Law of Cosines $c^2 = a^2 + b^2 - 2ab\cos C$ reduces to the Pythagorean theorem when:

When $C = 90^\circ$, $\cos 90^\circ = 0$, so the formula becomes $c^2 = a^2 + b^2 - 0 = a^2 + b^2$, which is exactly the Pythagorean theorem. The Law of Cosines is a generalization that works for all triangles.

Feature	Law of Sines	Law of Cosines
Formula	\(\frac{a}{\sin A} = \frac{b}{\sin B}\)	\(c^2 = a^2 + b^2 - 2ab\cos C\)
Best for	AAS, ASA, SSA	SAS, SSS
Requires	An angle-side opposite pair	Two sides + included angle, or all sides
Ambiguous case?	Yes (SSA)	No
Special case	Relates to circle circumradius	Reduces to Pythagorean theorem

Laws of Sines& Cosines

Why Do We Need New Tools?

Standard Triangle Labeling

What Can You Solve?

The Law of Sines

Proving the Law of Sines

The Law of Cosines

Three Forms of the Law of Cosines

When to Use Which Law?

Step-by-Step Problem Solving

Example 1: Law of Sines (AAS)

Example 2: Law of Cosines (SAS)

Example 3: Finding an Angle (SSS)

The Ambiguous Case (SSA)

Common Mistakes

Summary Comparison

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

Laws of Sines
& Cosines