Can the Law of Cosines find an obtuse angle?

Yes. If the cosine ratio is negative, the inverse cosine function will correctly return an obtuse angle (between 90 and 180 degrees).

Is there an ambiguous case for SSS triangles?

No. Unlike the Law of Sines (SSA), three side lengths uniquely define a single triangle, assuming they satisfy the triangle inequality theorem ($a+b>c$).

When should I switch to the Law of Sines?

Once you have one known angle-side opposite pair, the Law of Sines is computationally simpler. However, if finding a second angle, solving for the largest angle first using the Law of Cosines prevents the ambiguous case of the Law of Sines.

How to solve a triangle with the law of cosines

The Law of Cosines defines the relationship between the side lengths of a triangle and the cosine of its angles. It applies specifically to triangles where two sides and the included angle are known (SAS), or where all three side lengths are known (SSS).

The setup

For a triangle with sides $a$ , $b$ , and $c$ opposite angles $A$ , $B$ , and $C$ , the Law of Cosines states: $c^2 = a^2 + b^2 - 2ab \cos(C)$ . By permutation, $a^2 = b^2 + c^2 - 2bc \cos(A)$ and $b^2 = a^2 + c^2 - 2ac \cos(B)$ .

The steps

Identify the given triangle parameters as SAS (Side-Angle-Side) or SSS (Side-Side-Side). 2. Select the variation of the Law of Cosines that isolates the unknown side (for SAS) or the unknown angle (for SSS). 3. Substitute the known values and evaluate. 4. If finding an angle, isolate the cosine term and use the inverse cosine function. 5. Use the Law of Sines or repeat the Law of Cosines to find the remaining angles.

Checking the result

Verify that the sum of the angles equals exactly 180 degrees ( $A + B + C = 180^\circ$ ). Confirm that the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.

Common errors

Failing to take the square root of $c^2$ when solving for a side length. Executing calculations with the calculator set to radians instead of degrees. Subtracting $2ab$ from $a^2 + b^2$ before multiplying by $\cos(C)$ .

Worked example

Solve for the largest angle $C$ in a triangle with side lengths $a = 5$ , $b = 7$ , and $c = 9$ .

Identify the knowns: $a = 5$ , $b = 7$ , $c = 9$ . Choose the equation isolating angle $C$ : $c^2 = a^2 + b^2 - 2ab \cos(C)$ . Substitute values: $9^2 = 5^2 + 7^2 - 2(5)(7) \cos(C)$ . Evaluate squares and products: $81 = 25 + 49 - 70 \cos(C)$ . Simplify: $81 = 74 - 70 \cos(C)$ . Isolate the cosine term: $7 = -70 \cos(C)$ , so $\cos(C) = -0.1$ . Evaluate inverse cosine: $C = \cos^{-1}(-0.1)$ . Result: $C \approx 95.74^\circ$ .

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References: Stewart, J., Redlin, L., & Watson, S. (2015). Precalculus: Mathematics for Calculus. · OpenStax Algebra and Trigonometry, Chapter 10. · Khan Academy Unit: Trigonometric equations and identities.