Law of Sines

Q: When do you use the Law of Sines versus the Law of Cosines?

Use the Law of Sines when you know an angle and its opposite side plus at least one other angle or side (AAS, ASA, or SSA configurations). Use the Law of Cosines when you know all three sides (SSS) or two sides and the included angle (SAS). If no side-angle pair is known, the Law of Sines cannot be applied directly.

Q: What is the ambiguous case of the Law of Sines?

The ambiguous case occurs when you are given two sides and a non-included angle (SSA). Because the sine function gives the same value for an angle and its supplement (e.g., sin 40° = sin 140°), there may be zero, one, or two valid triangles. You must check whether each candidate angle produces a valid triangle whose angles sum to 180°.

Q: Does the Law of Sines work for right triangles?

Yes. The Law of Sines applies to all triangles — acute, obtuse, and right. For a right triangle with a 90° angle, sin 90° = 1, so the ratio simplifies: the hypotenuse equals the common ratio a/sin A. However, basic SOH-CAH-TOA is usually simpler for right triangles.

Law of Sines

Equations relating the sines of the interior angles of a triangle and the corresponding opposite sides.

Triangle ABC with sides a, b, c opposite to angles A, B, C. Formula: sin A/a = sin B/b = sin C/c

See also

Law of cosines

Key Formula

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

Where:

$a, b, c$ = The lengths of the three sides of the triangle
$A, B, C$ = The interior angles opposite to sides a, b, and c, respectively

Worked Example

Problem: In triangle ABC, angle A = 30°, angle B = 105°, and side a = 10. Find the length of side b.

Step 1: Find angle C using the fact that the angles of a triangle sum to 180°.

C = 180° - 30° - 105° = 45°

Step 2: Write the Law of Sines ratio relating sides a and b to their opposite angles.

\frac{a}{\sin A} = \frac{b}{\sin B}

Step 3: Substitute the known values into the equation.

\frac{10}{\sin 30°} = \frac{b}{\sin 105°}

Step 4: Evaluate the sines. Note that sin 30° = 0.5 and sin 105° ≈ 0.9659.

\frac{10}{0.5} = \frac{b}{0.9659}

Step 5: Solve for b by cross-multiplying.

b = 20 \times 0.9659 \approx 19.32

Answer: Side b ≈ 19.32

Another Example

This example demonstrates the ambiguous case (SSA), where two given sides and a non-included angle can produce two valid triangles. The first example had a unique solution (AAS), but here the inverse sine yields two possible angles.

Problem: In triangle ABC, angle A = 40°, side a = 12, and side b = 18. Find angle B. (This is the ambiguous case — SSA.)

Step 1: Write the Law of Sines to solve for sin B.

\frac{\sin B}{b} = \frac{\sin A}{a}

Step 2: Substitute the known values.

\frac{\sin B}{18} = \frac{\sin 40°}{12}

Step 3: Solve for sin B. Since sin 40° ≈ 0.6428:

\sin B = \frac{18 \times 0.6428}{12} = \frac{11.5704}{12} \approx 0.9642

Step 4: Find angle B using the inverse sine function. Since sin B ≈ 0.9642, there are potentially two solutions: B ≈ 74.6° or B ≈ 180° − 74.6° = 105.4°.

B \approx 74.6° \quad \text{or} \quad B \approx 105.4°

Step 5: Check both solutions. For B ≈ 74.6°, we get A + B = 40° + 74.6° = 114.6° < 180° ✓. For B ≈ 105.4°, we get A + B = 40° + 105.4° = 145.4° < 180° ✓. Both are valid, so two different triangles satisfy the given information.

B_1 \approx 74.6°, \quad B_2 \approx 105.4°

Answer: There are two valid solutions: B ≈ 74.6° or B ≈ 105.4°. This is the ambiguous case of the Law of Sines.

Frequently Asked Questions

When do you use the Law of Sines versus the Law of Cosines?

Use the Law of Sines when you know an angle and its opposite side plus at least one other angle or side (AAS, ASA, or SSA configurations). Use the Law of Cosines when you know all three sides (SSS) or two sides and the included angle (SAS). If no side-angle pair is known, the Law of Sines cannot be applied directly.

What is the ambiguous case of the Law of Sines?

The ambiguous case occurs when you are given two sides and a non-included angle (SSA). Because the sine function gives the same value for an angle and its supplement (e.g., sin 40° = sin 140°), there may be zero, one, or two valid triangles. You must check whether each candidate angle produces a valid triangle whose angles sum to 180°.

Does the Law of Sines work for right triangles?

Yes. The Law of Sines applies to all triangles — acute, obtuse, and right. For a right triangle with a 90° angle, sin 90° = 1, so the ratio simplifies: the hypotenuse equals the common ratio a/sin A. However, basic SOH-CAH-TOA is usually simpler for right triangles.

Law of Sines vs. Law of Cosines

	Law of Sines	Law of Cosines
Formula	a/sin A = b/sin B = c/sin C	c² = a² + b² − 2ab cos C
Known information needed	At least one side-angle opposite pair (AAS, ASA, SSA)	Three sides (SSS) or two sides and the included angle (SAS)
Ambiguous case	Yes — SSA can give 0, 1, or 2 solutions	No — always gives a unique solution
Best for finding	Unknown sides or angles when a side-angle pair is known	A missing side given SAS, or a missing angle given SSS
Complexity	Simpler — involves proportions	More involved — uses the cosine of an angle with squaring

Why It Matters

The Law of Sines is a core tool in trigonometry courses and appears on standardized tests including the SAT, ACT, and AP exams. It is essential in real-world applications such as surveying, navigation, and physics, where you need to find unknown distances or angles in non-right triangles. Mastering it also prepares you for more advanced topics like vectors, the Law of Cosines, and triangle area formulas involving sine.

Common Mistakes

Mistake: Pairing a side with the wrong angle — for example, using sin A with side b instead of side a.

Correction: Each side must be paired with the angle directly opposite to it. Side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. Always draw and label the triangle before setting up the ratio.

Mistake: Forgetting to check for the ambiguous case when given SSA, and only finding one angle from the inverse sine.

Correction: When you compute sin B = k (where 0 < k < 1), there are two candidate angles: B = sin⁻¹(k) and B = 180° − sin⁻¹(k). You must test both to see if each produces a valid triangle (all angles positive and summing to 180°).

Related Terms

Law of Cosines — Companion formula for solving non-right triangles
Sine — The trigonometric function used in the law
Triangle — The geometric figure the law applies to
Interior Angle — The angles referenced in the formula
Side of a Polygon — The sides referenced in the formula
Equation — The law is expressed as an equation
Corresponding — Each side corresponds to its opposite angle