How To Solve a Triangle | Master the Basics

Understanding how to solve a triangle is a fundamental skill in geometry and trigonometry, applicable across various fields from engineering to architecture. This process involves systematically determining all unknown sides and angles when some information about the triangle is already known, much like solving a puzzle with a clear set of rules.

What Does “Solving a Triangle” Mean?

Solving a triangle means determining the measures of all three angles and the lengths of all three sides. A triangle has six primary elements: three angles (A, B, C) and three sides (a, b, c), where side ‘a’ is opposite angle A, ‘b’ opposite B, and ‘c’ opposite C. To solve a triangle, you typically need at least three pieces of information, with at least one of them being a side length.

The type of information provided dictates the specific methods and formulas you will apply. Whether you are given side-angle-side (SAS), angle-side-angle (ASA), side-side-angle (SSA), side-side-side (SSS), or angle-angle-side (AAS), each scenario has a clear mathematical pathway.

Essential Tools: The Sum of Angles and Pythagorean Theorem

Before diving into more complex trigonometry, two foundational principles are indispensable for solving triangles, particularly right-angled ones.

Angle Sum Property

A universal truth for all Euclidean triangles is that the sum of their interior angles always equals 180 degrees. This property provides an immediate way to find a third angle if two angles are known. If you have angles A and B, angle C is simply 180° – (A + B). This principle holds true for both right and oblique (non-right) triangles.

Pythagorean Theorem

The Pythagorean Theorem is exclusively for right-angled triangles, which contain one angle measuring exactly 90 degrees. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (legs). Mathematically, this is expressed as a² + b² = c², where ‘c’ is the hypotenuse. This theorem allows you to find an unknown side length if the other two side lengths of a right triangle are known.

For more foundational mathematical concepts, resources like Khan Academy offer extensive explanations and practice problems.

Solving Right Triangles: SOH CAH TOA

When dealing with right-angled triangles, the primary tools for relating angles and side lengths are the trigonometric ratios: sine, cosine, and tangent. These ratios are often remembered by the mnemonic SOH CAH TOA.

Defining Sine, Cosine, Tangent

• Sine (SOH): The sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. (sin θ = Opposite / Hypotenuse)
• Cosine (CAH): The cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. (cos θ = Adjacent / Hypotenuse)
• Tangent (TOA): The tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. (tan θ = Opposite / Adjacent)

These ratios allow you to find an unknown side length if you know an angle and one side, or an unknown angle if you know two side lengths.

Using Inverse Functions

To find the measure of an angle when you know the ratio of two sides, you use the inverse trigonometric functions: arcsin (sin⁻¹), arccos (cos⁻¹), and arctan (tan⁻¹). For example, if you know the opposite side and the hypotenuse, calculating their ratio gives you the sine of the angle. Applying the arcsin function to this ratio will yield the angle’s measure.

Right Triangle Trigonometric Ratios (SOH CAH TOA)

Ratio	Formula	Application
Sine (sin)	Opposite / Hypotenuse	Find opposite side given angle and hypotenuse, or angle given opposite and hypotenuse.
Cosine (cos)	Adjacent / Hypotenuse	Find adjacent side given angle and hypotenuse, or angle given adjacent and hypotenuse.
Tangent (tan)	Opposite / Adjacent	Find opposite side given angle and adjacent, or angle given opposite and adjacent.

Solving Non-Right Triangles: Law of Sines

For triangles that do not contain a 90-degree angle (oblique triangles), the Pythagorean Theorem and SOH CAH TOA do not apply directly. Instead, we use the Law of Sines and the Law of Cosines. The Law of Sines states that the ratio of a side length to the sine of its opposite angle is constant for all three sides and angles in any triangle.

The formula is: a / sin A = b / sin B = c / sin C.

When to Apply the Law of Sines

The Law of Sines is particularly useful in two scenarios:

1. Angle-Side-Angle (ASA): When you know two angles and the included side.
2. Angle-Angle-Side (AAS): When you know two angles and a non-included side.
3. Side-Side-Angle (SSA): When you know two sides and a non-included angle. This case is known as the “ambiguous case” and requires careful consideration, as it can sometimes result in two possible triangles, one triangle, or no triangle at all.

To use the Law of Sines, you must have at least one complete pair of an angle and its opposite side. For instance, if you know side ‘a’ and angle A, you can use this ratio to find other unknown sides or angles if you have additional information.

For a detailed reference on trigonometric laws, Wolfram MathWorld provides comprehensive mathematical definitions.

Solving Non-Right Triangles: Law of Cosines

The Law of Cosines is a generalization of the Pythagorean Theorem, applicable to all triangles, including right triangles. It relates the lengths of the sides of a triangle to the cosine of one of its angles. There are three forms of the Law of Cosines, depending on which side or angle you are solving for:

• a² = b² + c² – 2bc cos A
• b² = a² + c² – 2ac cos B
• c² = a² + b² – 2ab cos C

When to Apply the Law of Cosines

The Law of Cosines is the preferred method in two specific cases:

1. Side-Side-Side (SSS): When you know the lengths of all three sides. You can use any form of the Law of Cosines to find one of the angles, then use the Angle Sum Property or Law of Sines to find the remaining angles.
2. Side-Angle-Side (SAS): When you know two side lengths and the included angle (the angle between the two known sides). You can use the appropriate form of the Law of Cosines to find the length of the third side, then proceed with the Law of Sines or Angle Sum Property to find the remaining angles.

The Law of Cosines is particularly robust because it does not suffer from the ambiguity that can arise with the Law of Sines in the SSA case, as it directly calculates side lengths or angles without requiring the sine of an angle, which has two possible values (acute and obtuse) for a given positive sine value.

Law of Sines vs. Law of Cosines Application Cases

Case	Law of Sines	Law of Cosines
ASA (Angle-Side-Angle)	Primary Method	Not applicable directly
AAS (Angle-Angle-Side)	Primary Method	Not applicable directly
SSA (Side-Side-Angle)	Primary Method (Ambiguous Case)	Can be used to resolve ambiguity or find missing side
SAS (Side-Angle-Side)	Not applicable directly	Primary Method
SSS (Side-Side-Side)	Not applicable directly	Primary Method

The Ambiguous Case (SSA)

The Side-Side-Angle (SSA) case is unique because it can sometimes lead to more than one possible triangle, or no triangle at all. This occurs when you are given two side lengths and a non-included angle. The ambiguity arises because the sine function has the same value for an angle and its supplement (e.g., sin 30° = sin 150°).

To navigate the ambiguous case, you typically use the Law of Sines to find the first unknown angle. Then, you must check for a second possible angle by considering its supplement. If both the original angle and its supplement, when combined with the known angle, result in a valid triangle (i.e., the sum of angles is less than 180°), then two triangles exist. If only one is valid, then one triangle exists. If neither is valid, or if the calculated sine value is greater than 1, then no triangle exists.

Careful calculation of the height of the triangle (h = b sin A, where A is the known angle and b is the adjacent known side) compared to the length of the opposite side (a) is key to determining the number of solutions.

A Systematic Approach to Solving Triangles

Solving a triangle efficiently often benefits from a structured approach. Begin by identifying the known information and the type of triangle (right or oblique).

1. Identify Knowns: List the given side lengths and angle measures.
2. Classify the Triangle: Determine if it’s a right triangle or an oblique triangle.
3. Choose the Right Tools:
   • For Right Triangles: Use the Angle Sum Property, Pythagorean Theorem, and SOH CAH TOA.
   • For Oblique Triangles: Use the Angle Sum Property, Law of Sines, and Law of Cosines.
4. Prioritize:
   • If you have SSS or SAS, start with the Law of Cosines to find a missing side or angle.
   • If you have ASA or AAS, start with the Law of Sines to find a missing side.
   • If you have SSA, use the Law of Sines, but be mindful of the ambiguous case.
5. Calculate Systematically: Find one unknown element at a time, using the most direct formula.
6. Verify: After finding all elements, ensure the Angle Sum Property holds (angles add to 180°), and side lengths are proportional to the sines of their opposite angles.

This systematic process ensures you apply the correct principles and minimize errors, guiding you through each step to fully solve the triangle.