ACT Math: Law of Cosines Solves Any Triangle (Right or Not)

Law of Cosines: a²=b²+c²-2bc·cos(A). Use it when you have: (1) two sides and the included angle (SAS), or (2) all three sides and need to find an angle (SSS). It works for any triangle, not just right triangles. If a triangle lacks a right angle, the Law of Cosines is your primary tool. The formula is an extension of the Pythagorean theorem; when the angle is 90°, cos(90°)=0, so it reduces to a²=b²+c².

Conceptually: The Law of Cosines accounts for how the angle between two sides affects the opposite side's length. A larger angle stretches the opposite side longer.

Type 1 (SAS—find missing side): A triangle has sides 5 and 7 with a 60° angle between them. Find the third side. a²=5²+7²-2(5)(7)cos(60°)=25+49-70(0.5)=74-35=39 → a=√39≈6.24. Type 2 (SSS—find missing angle): A triangle has sides 3, 4, and 5. Find angle C opposite the side of length 5. 5²=3²+4²-2(3)(4)cos(C) → 25=9+16-24cos(C) → 25=25-24cos(C) → 0=-24cos(C) → cos(C)=0 → C=90°. (This is a right triangle!) Type 3 (SAS with obtuse angle): Sides 8 and 6 with a 120° angle between them. Find third side. a²=8²+6²-2(8)(6)cos(120°)=64+36-96(-0.5)=100+48=148 → a=√148≈12.2. Memorize cos(0°)=1, cos(60°)=0.5, cos(90°)=0, cos(120°)=-0.5, cos(180°)=-1 to speed calculations.

Practice these three types daily until you solve each in under 2 minutes.

Problem 1: Sides 7 and 9, angle 45° between them. Find the third side. a²=49+81-2(7)(9)cos(45°)=130-126(0.707)≈20.9 → a≈4.6. Problem 2: Sides 5, 6, 8. Find the largest angle (opposite the longest side, 8). 8²=5²+6²-2(5)(6)cos(C) → 64=25+36-60cos(C) → 64=61-60cos(C) → 3=-60cos(C) → cos(C)=-0.05 → C≈93°. Problem 3: Sides 10 and 12, 30° angle. Find third side. a²=100+144-2(10)(12)cos(30°)=244-240(0.866)≈33.8 → a≈5.8. Problem 4: Sides 6, 8, 10. Find angle C opposite the side of 10. 10²=6²+8²-2(6)(8)cos(C) → 100=100-96cos(C) → 0=-96cos(C) → C=90°. Complete all four daily for one week until speed and accuracy are automatic.

On test day, write the Law of Cosines formula and common cos values on your scratch paper immediately.

Law of Cosines questions appear in 1-2 ACT Math sections and are often considered "hard" by weaker test-takers. However, once you know the formula and a few common cosine values, solving these questions is mechanical. A student who knows the Law of Cosines answers these confidently while many others guess or leave them blank. Each correct Law of Cosines question is a point advantage over the competition.

Invest one week in this. By test day, non-right triangle problems become just another calculation.