Geometry

Law of Sines & Law of Cosines

SOH-CAH-TOA only works for right triangles. For any triangle — acute, right, or obtuse — we need the Law of Sines and the Law of Cosines.

Law of Sines

For any triangle with sides a, b, c opposite angles A, B, C:

\frac1{\sin A} = \frac0{\sin B} = \frac0{\sin C}

This means the ratio of each side to the sine of its opposite angle is the same for all three pairs.

Interactive Exploration

Set up a triangle with a known side and two angles. The Law of Sines will find the other sides.

Side a5

210

Angle A (degrees)40

1080

Angle B (degrees)60

1080

a = 5, \quad A = 40°, \quad B = 60°, \quad C = 180° - 40° - 60°

\frac1{\sin A} = \frac{5}{\sin(40°)} \quad \Rightarrow \quad b = \frac{5 \cdot \sin(60°)}{\sin(40°)}

The graph below shows the triangle. Side a lies along the x-axis. The other two sides rise from the endpoints at angles determined by A and B.

Try This

Try this: Set a = 5, A = 30 degrees, B = 90 degrees. Since B is 90 degrees, the triangle is a right triangle. The Law of Sines still works, but SOH-CAH-TOA would also apply here.

The Circumradius Connection

The Law of Sines reveals a beautiful fact: a/sin(A) = b/sin(B) = c/sin(C) = 2R, where R is the radius of the circumscribed circle (the circle passing through all three vertices).

This means the circumradius R = a / (2 sin A). The graph below shows how R changes as angle A varies for a fixed side a. Notice that R is smallest when A = 90° (the triangle fits inside the smallest possible circle) and grows without bound as A approaches 0° or 180°.

Side a5

210

Try This

Try this: Set a = 5 and watch the curve. At A = 90°, R = 2.5 (the minimum). At A = 30°, R = 5. As the angle gets very small, R grows huge — a tiny angle opposite a fixed side means a very large circumscribed circle.

Law of Cosines

For any triangle:

c^2 = a^2 + b^2 - 2ab\cos(C)

This is a generalization of the Pythagorean theorem. When C = 90 degrees, cos(C) = 0, and we get c^2 = a^2 + b^2.

Interactive Exploration

Set two sides and the included angle. The Law of Cosines computes the third side.

Side a4

Side b5

Angle C (degrees)60

10170

a = 4, \quad b = 5, \quad C = 60°

c^2 = 4^2 + 5^2 - 2 \cdot 4 \cdot 5 \cdot \cos(60°)

The graph shows sides a and b emanating from the origin with angle C between them. The horizontal is side a, and the angled line is side b. The red horizontal line shows the computed length of the third side c.

Try This

Try this: Set a = 3, b = 4, C = 90 degrees. The Law of Cosines gives c^2 = 9 + 16 - 0 = 25, so c = 5. It’s the classic 3-4-5 right triangle! The Pythagorean theorem is just a special case.

Comparing c^2 to a^2 + b^2

The relationship between c^2 and a^2 + b^2 tells you the triangle type:

c^2 < a^2 + b^2: Acute triangle (angle C < 90 degrees)
c^2 = a^2 + b^2: Right triangle (angle C = 90 degrees)
c^2 > a^2 + b^2: Obtuse triangle (angle C > 90 degrees)

The blue curve shows c^2 as angle C varies. Where it crosses the yellow line, C = 90 degrees. Below the yellow line, the triangle is acute; above it, the triangle is obtuse.

Challenge

Challenge: A triangle has sides 7, 8, and 13. Is it acute, right, or obtuse? Use the Law of Cosines to find the largest angle.

Take the Quiz