Given an Angle and Three Sides- Triangle Solving Techniques

What "Angle and Three Sides" Actually Means

Let's get something straight first. You can't have exactly "an angle and three sides" — that's four pieces of information. What you probably mean is one of these real scenarios:

• SSS — You know all three sides, need to find the angles
• SAS — You know two sides and the angle between them
• SSA — You know two sides and an angle that isn't between them (the tricky one)

Each scenario requires a different approach. Here's how to handle every case.

SSS: Finding Angles When You Know All Three Sides

When you have all three sides and need to find the angles, Law of Cosines is your only real option. There's no shortcut.

The formula:

c² = a² + b² - 2ab·cos(C)

Rearrange it to solve for any angle:

cos(C) = (a² + b² - c²) / 2ab

Step-by-Step Process

1. Label your sides a, b, c (it doesn't matter which is which, but c is typically opposite the angle you're solving for)
2. Pick the largest side — this angle will be the largest and is usually easiest to find first
3. Plug into the Law of Cosines formula
4. Use inverse cosine to get your angle in degrees
5. Repeat for the remaining angles

Worked Example

Given sides a = 7, b = 9, c = 5

Find angle C (opposite side c):

cos(C) = (7² + 9² - 5²) / (2 × 7 × 9)

cos(C) = (49 + 81 - 25) / 126

cos(C) = 105 / 126 = 0.8333

C = cos⁻¹(0.8333) = 33.3°

Repeat the process for the other angles. That's it. No magic here — just arithmetic.

SAS: Two Sides and the Included Angle

This one's straightforward. You have two sides and the angle between them. You can find the third side immediately using Law of Cosines, then work backward to find the remaining angles.

Step-by-Step Process

1. Use Law of Cosines to find the unknown side opposite the given angle
2. Now you have SSS — return to the previous section
3. Find the remaining angles using Law of Cosines

The math:

c² = a² + b² - 2ab·cos(C)

Where C is your known angle. Solve for c. Then you're in SSS territory.

SSA: The Ambiguous Case (This Is Where It Gets Weird)

Two sides and an angle that isn't between them. This is called the ambiguous case because you might get zero, one, or two valid triangles from the same information.

Here's why: when you use Law of Sines to find a missing angle, sine values repeat. sin(30°) = sin(150°). Both are valid angles in different triangles.

The Height Check (How to Know What You're Dealing With)

Before you do any serious math, check the height from the known angle to the known side:

h = b·sin(A)

Where A is your known angle and b is the side adjacent to it.

Now compare this height (h) to your known third side (a):

• If a < h: No triangle exists. The side is too short to reach.
• If a = h: Exactly one right triangle exists.
• If h < a < b: Two triangles exist.
• If a ≥ b: Exactly one triangle exists.

Step-by-Step for SSA

1. Check the height as shown above
2. Use Law of Sines to find sin(B): sin(B)/b = sin(A)/a
3. If sin(B) > 1: No solution. Stop.
4. If sin(B) = 1: One right triangle. Calculate B = 90°.
5. If sin(B) < 1: Calculate the first solution — B = sin⁻¹(value)
6. Calculate the second possible angle: B₂ = 180° - B₁
7. Check if this creates a valid triangle (angles must sum to 180°)
8. If valid, calculate the third angle and remaining side

Worked Example

Given: A = 40°, a = 8, b = 10

Height check: h = 10 × sin(40°) = 10 × 0.643 = 6.43

Since a = 8 > h = 6.43, and a < b, we expect two possible triangles.

Using Law of Sines:

sin(B) / 10 = sin(40°) / 8

sin(B) = 10 × 0.643 / 8 = 0.804

B₁ = sin⁻¹(0.804) = 53.4°

B₂ = 180° - 53.4° = 126.6°

Triangle 1: A = 40°, B = 53.4°, C = 180° - 93.4° = 86.6°

Triangle 2: A = 40°, B = 126.6°, C = 180° - 166.6° = 13.4°

Both are valid. That's the ambiguous case.

Quick Reference: Which Method Do I Use?

Law of Sines vs Law of Cosines — When to Use Which

Use Law of Cosines when:

• You have SSS (all three sides)
• You have SAS (two sides and the angle between them)
• You need to find a side, not an angle

Use Law of Sines when:

• You have AAS or ASA (angles and at least one side)
• You have SSA but only one triangle is possible
• You need to find an angle, not a side

The general rule: Law of Cosines for SSS/SAS, Law of Sines for AAS/ASA. SSA is the exception — use Law of Sines but expect complications.

Common Mistakes That Ruin Your Answers

• Forgetting to check the ambiguous case — SSA isn't always a single solution. Always check the height first.
• Rounding too early — Keep full decimal precision until the final answer. Rounding mid-calculation compounds errors.
• Using degrees when your calculator is in radians — Check your mode. This destroys every answer.
• Assuming the largest angle is opposite the largest side — It's true, but verify it. If the largest side is shorter than the sum of the other two, the triangle exists.

How To Actually Solve These Problems

Step 1: Identify Your Case

Write down exactly what you know. Label sides a, b, c and angles A, B, C. Match up what you have with SSS, SAS, or SSA.

Step 2: Choose Your Tool

SSS or SAS → Law of Cosines first. AAS/ASA → Law of Sines. SSA → Height check, then Law of Sines.

Step 3: Solve the First Unknown

Find the easiest piece first. Usually the side opposite a known angle, or the largest angle in SSS problems.

Step 4: Work Toward the Remaining Pieces

Once you have three pieces that match SSS or SAS, you're in familiar territory. Repeat the process.

Step 5: Verify

Check that all three angles sum to 180°. Verify that sides are proportional to sines of opposite angles (Law of Sines holds). If these don't check out, you made an arithmetic error somewhere.

The Brutal Truth About Triangle Solving

There are no tricks here. You either know the formulas and how to apply them, or you don't. Law of Cosines and Law of Sines are the only tools you need — everything else is just deciding which one to use and how many solutions to expect.

The ambiguous case trips up almost everyone. The only way to handle it is to always check the height first when you have SSA. That's not optional — it's how you know whether to expect one triangle, two, or none.

Get the formulas right. Check your calculator mode. Verify your answers. That's the whole process.