ACT Math: Law of Cosines and Law of Sines for Non-Right Triangles

Law of Sines: a/sin(A)=b/sin(B)=c/sin(C). Use when you know: (1) An angle and the side opposite it, plus another angle or side (AAS, ASA, SSA). Law of Cosines: c²=a²+b²-2ab·cos(C). Use when you know: (1) Two sides and the included angle (SAS), or (2) All three sides (SSS). On the ACT, you'll rarely need both in one problem, but you must recognize which setup applies. Example problem 1: "A triangle has sides 5 and 7, and the angle between them is 60°. Find the third side." This is SAS → use Law of Cosines: c²=5²+7²-2(5)(7)cos(60°). Example problem 2: "A triangle has angle A=40°, side a=10, and angle B=65°. Find side b." This is AAS → use Law of Sines: 10/sin(40°)=b/sin(65°). Identify which sides/angles you know, then pick the law that matches.

Mnemonic: Sines are for angles and opposite sides. Cosines include the angle between two known sides.

Error 1: Using Law of Cosines when Law of Sines is simpler (wastes time, risk arithmetic errors). Error 2: Misidentifying which angle is "included" (the angle between two sides). Error 3: Forgetting that inverse trig functions (sin⁻¹, cos⁻¹) give one answer, but some problems have two valid triangles (SSA ambiguity). Error 4: Setting up the equation correctly but mixing up which side is opposite which angle. Error 5: Using a calculator in the wrong mode (degree vs radian). Before you start: (1) Identify the known sides/angles. (2) Match to SAS, SSA, AAS, ASA, or SSS. (3) Pick the law. (4) Set up the equation carefully. (5) Check that your answer makes sense (angles sum to 180°).

Verification: After solving, verify your triangle is valid. Sum angles: A+B+C=180°. Check: Opposite sides and angles have proportional relationships (larger angle opposite larger side).

Problem 1: Sides a=8, b=10, included angle C=45°. Find side c. Use Law of Cosines: c²=64+100-2(8)(10)cos(45°)≈42.27, so c≈6.5. Problem 2: Angle A=50°, side a=7, angle B=70°. Find side b. Use Law of Sines: 7/sin(50°)=b/sin(70°). Solve: b≈8.7. Problem 3: Sides a=5, b=6, c=7. Find angle C. Use Law of Cosines: 49=25+36-2(5)(6)cos(C). Solve: cos(C)=0.1, so C≈84°. For each, check: Do angles sum to 180°? Is the largest side opposite the largest angle?

Daily practice: Solve one non-right triangle problem daily. Alternate between Law of Sines and Law of Cosines. Verify your answer makes geometric sense.

Non-right triangle questions appear 1-2 times per ACT Math section, usually in the medium difficulty range. These are worth the same points as easier questions but feel harder to many students, so if you master the laws, you're taking advantage of a high-value opportunity. Most students either avoid these questions or set them up wrong. Knowing when and how to apply Law of Sines and Law of Cosines puts you ahead and gives you quick, reliable points.

Spend 2-3 days drilling these laws. By test day, you'll recognize a non-right triangle problem and solve it confidently while others skip it.