How To Find Angle Of Triangle Given 2 Sides

How to Find an Angle in a Triangle Given Two Sides: A Complete Guide

Determining the unknown angles of a triangle when you only know the lengths of two sides is a fundamental challenge in geometry and trigonometry. This scenario, often arising in construction, navigation, and physics, requires moving beyond simple right-triangle ratios. The solution depends critically on what other information you have. Are you dealing with a right triangle, or is it an oblique triangle? Do you know the length of the third side, or perhaps another angle? This guide will systematically unpack the methods, from the simplest to the more complex, ensuring you can confidently find any missing angle with just two side lengths and the appropriate additional piece of data.

Foundational Concepts: The Trigonometric Toolkit

Before applying specific formulas, you must understand the core principles that govern all triangles.

The Right Triangle Baseline: SOHCAHTOA

If your triangle is a right triangle (one 90° angle), the process is straightforward using the primary trigonometric ratios—sine, cosine, and tangent—memorized via the mnemonic SOHCAHTOA:

• Sine = Opposite / Hypotenuse
• Cosine = Adjacent / Hypotenuse
• Tangent = Opposite / Adjacent

Here, the two known sides must include the hypotenuse (the side opposite the right angle) and one leg. You directly apply the appropriate ratio to find the acute angle. For example, if you know the adjacent side and the hypotenuse, use cosine (cos(θ) = adjacent/hypotenuse).

The Laws for All Triangles: Sine and Cosine

For non-right triangles (oblique triangles), the universal tools are the Law of Sines and the Law of Cosines.

• Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) This law relates each side to the sine of its opposite angle. It is most powerful when you know two angles and one side (AAS or ASA) or two sides and an angle opposite one of them (SSA).
• Law of Cosines: c² = a² + b² - 2ab*cos(C) This is a generalization of the Pythagorean Theorem. It is essential when you know all three sides (SSS) or two sides and the included angle (SAS). You can rearrange it to solve for an angle: cos(C) = (a² + b² - c²) / (2ab).

Step-by-Step Methods Based on Your Given Information

Your path forward is determined by the specific combination of known elements. Let's break it down.

Scenario 1: Right Triangle (Known: Two Sides, One is the Hypotenuse)

This is the simplest case. Identify the right angle (often given or implied by a square corner symbol).

1. Label the sides relative to the unknown angle θ: Opposite (across from θ), Adjacent (next to θ, not the hypotenuse), Hypotenuse (opposite the right angle).
2. Choose the correct ratio from SOHCAHTOA based on which two sides you know.
3. Set up the equation and solve for θ using the inverse trigonometric function on your calculator (sin⁻¹, cos⁻¹, tan⁻¹).
4. Example: In a right triangle, the side adjacent to angle A is 4 cm, and the hypotenuse is 5 cm. cos(A) = 4/5 = 0.8. Therefore, A = cos⁻¹(0.8) ≈ 36.87°.

Scenario 2: Oblique Triangle - SAS (Known: Two Sides and the Included Angle)

You know two sides and the angle between them.

1. You have sides a and b, and the included angle C.
2. Use the Law of Cosines to find the third side c first: c² = a² + b² - 2ab*cos(C).
3. Now you have all three sides (a, b, c). Switch to the Law of Sines to find one of the unknown angles. It’s easiest to find the angle opposite the newly calculated side or the smaller known side to avoid ambiguity.
4. Example: Sides a=7, b=10, included angle C=30°. Find side c: c² = 7² + 10² - 2*7*10*cos(30°) ≈ 49 + 100 - 140*0.866 ≈ 149 - 121.24 = 27.76. So c ≈ √27.76 ≈ 5.27. Now use Law of Sines: sin(A)/7 = sin(30°)/5.27. sin(A) = 7 * 0.5 / 5.27 ≈ 0.664. A = sin⁻¹(0.664) ≈ 41.6°. Finally, find angle B: B = 180° - 30° - 41.6° = 108.4°.

Scenario 3: Oblique Triangle - SSS (Known: All Three Sides)

You know the length of all three sides (a, b, c).

1. Use the rearranged Law of Cosines to find the largest angle first (opposite the longest side). This is a best practice to ensure accuracy and avoid the ambiguous case later.
2. cos(C) = (a² + b² - c²) / (2ab). Calculate c² and the denominator, then find cos(C), and finally C = cos⁻¹(...).
3. With one angle known, use the Law of Sines to find a second angle.
4. Use the Triangle Sum Theorem (all angles sum to 180°) for the third angle.
5. Example: Sides a=5, b=7, c=9. Longest side is c=9, so find angle C first. cos(C) = (5² + 7² - 9²) / (2*5*7) = (25 + 49 - 81) / 70 = (-7) / 70 = -0.1. `C = cos⁻¹(-0