Law of Sines vs Law of Cosines: Which One to Use
Pick the right law every time: sines for AAS, ASA and SSA; cosines for SAS and SSS. A decision chart, the cosine formula, and one example of each case.
You Have Three Known Parts and Need the Rest
You stare at a triangle that is not a right triangle. You have three measurements, some sides, some angles, and you need the missing ones. The choice between the law of sines vs law of cosines determines whether you get an answer in two steps or twenty minutes of dead ends. The rule is simple: match your given information to the correct law, and the rest follows.
Decision Chart by Given Information
Your given information type tells you which law to use first. The law of sines works when you know two angles and any side (AAS or ASA). The law of cosines works when you know two sides and the included angle (SAS) or all three sides (SSS). The ambiguous case, two sides and a non-included angle (SSA), starts with the law of sines but requires a check for 0, 1, or 2 possible triangles.
If you have two angles, find the third angle by subtracting from 180°, then use the law of sines. If you have two sides and the included angle, use the law of cosines to find the third side, then the law of sines for the remaining angles. If you have three sides, use the law of cosines to find any angle, then the law of sines for the rest. If you have two sides and a non-included angle, begin with the law of sines and immediately check the ambiguous case rules.
| Given Information | Start With | Typical Steps |
|---|---|---|
| Two angles and a side (AAS or ASA) | Law of sines | Find third angle, then one side via sine ratio, then final side |
| Two sides and included angle (SAS) | Law of cosines | Find third side, then use law of sines for angles |
| Three sides (SSS) | Law of cosines | Find one angle, then use law of sines for remaining two |
| Two sides and non-included angle (SSA) | Law of sines, then check ambiguous case | Calculate possible angle, then check for 0, 1, or 2 triangles |
| Right triangle | Standard sine/cosine ratios | Use SOH-CAH-TOA instead |
When to Use Law of Sines
The law of sines states that in any triangle, the ratio of a side length to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C) = 2R, where R is the circumradius. Use the law of sines when you have a pair of known opposite side and angle, that is, you know an angle and the side opposite it, plus one more piece of information.
AAS and ASA Problems
For AAS problems (two angles and a non-included side), you already have the third angle from the 180° sum. You then set up a proportion to find the unknown side. For ASA problems (two angles and the included side), you first find the third angle and then use the law of sines to find the other two sides. The law of sines also handles the ambiguous SSA case, but only after you apply the case table.
Law of Cosines Formula and Forms
The law of cosines extends the Pythagorean theorem to any triangle. For a triangle with sides a, b, c and opposite angles A, B, C, the three forms are: a² = b² + c², 2bc cos(A), b² = a² + c², 2ac cos(B), c² = a² + b², 2ab cos(C). The pattern is always (side you want to find) squared equals the sum of the squares of the other two sides minus twice the product of those two sides times the cosine of the included angle.
Use the law of cosines for SAS problems (two sides and the included angle) and for SSS problems (three sides). For SAS, you solve for the unknown side first, then use the law of sines to find the remaining angles. For SSS, you solve for any angle first, then use the law of sines for the other two angles.
Converting Formulas for Your Problem
To find side a when you know sides b and c and angle A (the angle opposite a), use a² = b² + c², 2bc cos(A). To find angle A when you know all three sides, rearrange to cos(A) = (b² + c², a²)/(2bc). The law of cosines always gives a unique angle between 0° and 180° because the cosine function is one-to-one in that range, unlike the inverse sine which can return both acute and obtuse possibilities.
SAS Example: Two Sides and the Included Angle
Triangle ABC has side b = 8, side c = 5, and angle A = 50° (the angle between sides b and c). Find side a and angles B and C.
Step 1: Use the law of cosines to find side a. a² = b² + c² - 2bc cos(A) = 8² + 5² - 2(8)(5)cos(50°) = 64 + 25 - 80(0.6428) = 89 - 51.42 = 37.58. So a ≈ 6.13.
Step 2: Use the law of sines to find angle B. a/sin(A) = b/sin(B) gives 6.13/sin(50°) = 8/sin(B). sin(50°) = 0.7660, so 6.13/0.7660 = 8/sin(B). The common ratio is 8.00, so sin(B) = 8/8.00 = 1.00, so B = 90°.
Step 3: Find angle C = 180° - 50° - 90° = 40°. The triangle is solved. Notice the law of cosines gave a unique side, and the law of sines then found the obtuse angle correctly because the ratio was exact.
SSS Example: Three Sides
Triangle ABC has sides a = 7, b = 9, c = 11. Find all three angles.
Step 1: Use the law of cosines to find the largest angle first (opposite the longest side, c = 11). cos(C) = (a² + b², c²)/(2ab) = (49 + 81-121)/(2*7*9) = (9)/(126) = 0.0714. So C = arccos(0.0714) ≈ 85.9°.
Step 2: Use the law of sines to find angle A. a/sin(A) = c/sin(C) gives 7/sin(A) = 11/sin(85.9°). sin(85.9°) ≈ 0.997, so 7/sin(A) = 11/0.997 ≈ 11.03. Thus sin(A) = 7/11.03 ≈ 0.635, so A ≈ 39.4°.
Step 3: Find angle B = 180°, 85.9°, 39.4° = 54.7°. The law of cosines gave the first angle uniquely, avoiding any ambiguity. Using law of sines for the other angles works because you know the largest angle and thus the other two must be acute.
Mixing Both Laws in One Problem
Some problems require switching between the two laws. The most common scenario: you start with an SSA problem that has two solutions. The law of sines gives you a first possible angle, but you must check whether a second angle (180° minus the first) also works. For each valid angle, you find the third angle, then use the law of sines again to find the remaining side. The law of cosines can verify your results, if the computed sides and angles satisfy the law of cosines, your solution is correct.
SSA Example with Two Solutions
In triangle ABC, side a = 10, side b = 12, and angle A = 40° (SSA). Use the law of sines: 10/sin(40°) = 12/sin(B). sin(40°) ≈ 0.643, so the common ratio is 10/0.643 ≈ 15.55. Then sin(B) = 12/15.55 ≈ 0.772, so B ≈ 50.5°. The second possible angle is 180° - 50.5° = 129.5°. Check if both work: for B = 50.5°, C = 180° - 40° - 50.5° = 89.5°, and side c = 15.55 * sin(89.5°) ≈ 15.55. For B = 129.5°, C = 180° - 40° - 129.5° = 10.Both triangles are valid because the sides satisfy the triangle inequality. The law of cosines can verify: for the first triangle, check c² = a² + b² - 2ab cos(C) using a=10, b=12, C=89.6, matching the law of sines result.
When to Use Law of Cosines
Use the law of cosines whenever the given information does not include a pair of opposite side and angle. That means SAS and SSS problems should always start with the law of cosines. The law of sines cannot solve SAS because you do not have an angle opposite a known side. For SSS, the law of sines requires at least one known angle, which you do not have.
The law of cosines formula c² = a² + b², 2ab cos(C) is the key. Learn it in its three forms and you can solve any SAS or SSS triangle in one step. The failure case: students try to use the law of sines for SAS and get stuck because they have no opposite pair. If you have two sides and the included angle, start with the law of cosines and you will never get stuck.
What Most Often Goes Wrong
The single thing that most often goes wrong is the SSA ambiguous case. Students apply the law of sines, get one angle from the inverse sine, and stop. They do not check for a second possible angle. The calculator returns only the acute angle, and if the problem has two solutions, one valid triangle is lost. The fix is always to compute both possible angles after the first sine result and check each against the triangle sum and side length conditions. The law of sines works for any oblique triangle, but it demands that you apply the full ambiguous case rules every time you have SSA input.
Common Questions
Can I use the law of sines if I only know two sides and the included angle?
No. The law of sines requires a known opposite side-angle pair. SAS gives you two sides and the angle between them, not an angle opposite a known side. Start with the law of cosines.
How do I know if my SSA problem has two solutions without drawing?
Apply the ambiguous case table. Given angle A and sides a (opposite A) and b (adjacent to A): if A is acute, a < b sin(A) gives 0 solutions; a = b sin(A) gives 1 right triangle; b sin(A) < a < b gives 2 solutions; a ≥ b gives 1 solution. If A is obtuse, a ≤ b gives 0 solutions; a > b gives 1 solution.
What does the law of cosines formula look like for side b?
b² = a² + c², 2ac cos(B). The pattern is always (side squared) = (sum of squares of other two sides)-2(product of other two sides)(cosine of opposite angle).
When should I use the law of sines instead of the law of cosines?
Use the law of sines for AAS, ASA, and SSA problems. Use the law of cosines for SAS and SSS problems. The law of sines is faster when you have an opposite side-angle pair; the law of cosines is the fallback when you do not.
Why does the law of sines sometimes give a negative angle?
The inverse sine function on a calculator returns only acute angles (between, 90° and 90°). If the true angle is obtuse (between 90° and 180°), you must subtract the acute result from 180°. The law of cosines gives the correct obtuse angle directly without this step.
What is the circumradius R in the law of sines formula?
R is the radius of the circumcircle that passes through all three vertices. The formula a/sin(A) = 2R shows that the common ratio equals twice the circumradius. In practice, you do not need R to solve triangles; it is a derivation tool.
How do I verify my law of sines result?
Use the law of cosines to check one of the computed sides or angles. If the computed values satisfy the cosine law for one combination, the solution is correct. This catches rounding errors and missed second solutions.