Sine Law and Cosine Law

The Sine Law states that the ratio of the side length to the sine of the opposite angle is the same for all sides: a/sin A = b/sin B = c/sin C. It's perfect when you have a mix of angles and sides.

The Cosine Law connects all three sides with any angle: c² = a² + b² - 2ab cos C. Think of it as an extension of the Pythagorean theorem for triangles that aren't right triangles.

When given two angles and one side (Case I), you can easily find the third angle by remembering that the sum of angles in a triangle equals 180°. Then use the Sine Law to find the remaining sides.

💡 Always check your work: In the example where a = 20, A = 50°, and B = 70°, we first found C = 60°, then used the Sine Law twice to find c ≈ 22.61 and b ≈ 24.53.

The Ambiguous Case

When given two sides and an angle opposite one of them (Case II), you might find zero, one, or two possible triangles! This is called the ambiguous case.

Before diving into calculations, check if the triangle is possible by comparing the opposite side to the product of the adjacent side and sine of the angle (a ≥ b·sin A).

For example, with a = 20, b = 25, and A = 40°, we check: 20 ≥ 25·sin 40° (which is 16.07). Since this is true, we have two possible solutions.

To solve, use the Sine Law to find angle B, which gives us two possibilities. Then find angle C using the fact that angles in a triangle sum to 180°, and finally calculate side c using the Sine Law again.

💡 The ambiguous case is the only situation where one set of measurements can produce two different triangles, so be sure to check all possible solutions!

Right Triangles and Two Sides + Included Angle

When one solution is a right triangle (90°), you can verify it by checking if the sides follow the Pythagorean theorem $c² = a² + b²$.

For a right triangle with b = 20, c = 40, and A = 30°, we can find the third side using the Pythagorean theorem: a = √$c² - b²$ = 34.64. The angles must sum to 180°, so we get A = 60°, B = 30°, and C = 90°.

When given two sides and the included angle (Case III), the Cosine Law is your best friend. For example, with a = 25, b = 32, and C = 75°, first find the third side: c² = a² + b² - 2ab·cos C.

After finding the third side, use the Cosine Law again to find the remaining angles: cos A = $b² + c² - a²$/(2bc) and cos B = $a² + c² - b²$/(2ac).

💡 The included angle is the one formed by the two given sides—using the wrong angle in the Cosine Law will give you incorrect results!

Solving Triangles with Three Sides

When all three sides of a triangle are known (Case IV), you can find each angle using the Cosine Law rearranged for angles.

For a triangle with a = 20, b = 28, and c = 36, we use these formulas:

• cos A = $b² + c² - a²$/(2bc) = 0.85573, so A ≈ 31.16°
• cos B = $c² + a² - b²$/(2ac) = 0.50704, so B ≈ 59.53°
• cos C = $a² + b² - c²$/(2ab) = -0.09739, so C ≈ 95.59°

These calculations show how the Cosine Law connects the sides of a triangle to its angles. Just plug in the values and solve for each angle.

💡 Always check your work by making sure the three angles sum to 180°. If they don't, you've made a calculation error somewhere!