Law of Cosines: How to Solve Oblique Triangles Without Guessing on the FE Exam

The entire Law of Cosines setup hinges on one decision: matching the side you’re solving for to the angle in the cosine term.

Get that right—side c opposite angle C—and the rest is arithmetic.

Mismatch them, and you’ll calculate an answer that’s 30% too large or too small, and none of the answer choices will be close.

That matching rule sounds simple until you’re holding a triangle with sides a, b, c and angles A, B, C, and you’re trying to remember whether the formula is c² = a² + b² – 2ab cos(C) or if the minus sign flips to a plus depending on the angle.

You know it’s not the Pythagorean theorem because there’s no right angle, but you’re not sure if you’re one sign error away from a wrong answer.

Under exam pressure, that uncertainty turns a 90-second problem into a 3-minute problem because you’re second-guessing the setup instead of trusting the structure.

This guide walks you through a systematic approach that works on any Law of Cosines problem the FE throws at you—from basic side-angle-side setups where you’re solving for the third side, to problems that give you all three sides and ask you to find an angle.

You’ll learn exactly how to match sides to angles without confusion, when to use the Law of Cosines versus other triangle tools, and how to set up the equation correctly on the first try so you’re not wasting time rechecking your work.

Before we walk through it step by step, watch this short video.

It shows you the full process from labeling the triangle to matching sides and angles to executing the calculation without sign errors.

You’ll see exactly where students typically lose time on oblique triangle problems—mismatching labels or forgetting the minus sign—and how the workflow prevents both mistakes completely.

What Is the Law of Cosines?

The Law of Cosines is a relationship that connects the three sides of any triangle to one of its angles.

For any triangle with sides a, b, and c, and angle C opposite side c, the relationship is:

c² = a² + b² – 2ab cos(C)

When the angle C is 90 degrees, the cosine term becomes zero, and the equation collapses into the Pythagorean theorem.

When the angle isn’t 90 degrees, the correction term adjusts the relationship so the equation works for all triangles.

This means the Law of Cosines is the tool you reach for when the Pythagorean theorem doesn’t apply—when the triangle doesn’t have a right angle, but you still need to find an unknown side or angle.

Think of it like this: the Pythagorean theorem is the special case. The Law of Cosines is the general tool that works everywhere.

If you tried to force the Pythagorean theorem onto an oblique triangle, your answer would be wrong because you’d be ignoring the angle.

The Law of Cosines corrects for that angle and gives you the real relationship between the sides.

On the FE Exam, Law of Cosines problems show up when you’re given a triangle that isn’t a right triangle and you need to find a missing side or angle.

The problem will typically give you two sides and the angle between them, or it will give you all three sides and ask you to find an angle.

The moment you see an oblique triangle with that kind of setup, you know the Law of Cosines is the move.

The key insight: the side you’re solving for is always opposite the angle in the formula.

If you’re solving for side c, angle C is in the equation.

If you’re solving for angle A, side a is in the equation.

That’s the matching rule that keeps the setup clean.

Breaking Down Law of Cosines Step by Step

On the FE, you’ll write c² = a² + b² – 2ab cos(C) correctly about 70% of the time.

The other 30%, you’ll either flip the minus sign to a plus, or you’ll match side a with angle B instead of angle A, and your answer will miss every choice by 20-30%.

The workflow below eliminates that 30% by forcing you to label the triangle first and verify the side-angle match before you substitute anything.

Let’s lay out the steps.

Step 1: Identify What You’re Given and Label the Triangle

The first thing you need to do is read the problem carefully and figure out what configuration you’re working with.

Draw the triangle if one isn’t given, and label each side and angle with the letters a, b, c and A, B, C.

The matching rule is simple: lowercase letters are sides, uppercase letters are angles, and side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.

This isn’t arbitrary labeling—it’s how the Law of Cosines formula is structured, and if you mismatch sides and angles, your setup breaks.

Once you’ve labeled the triangle, identify what the problem gives you:

If you know two sides and the angle between them (SAS), you’re solving for the third side.

If you know all three sides (SSS), you’re solving for one of the angles.

Write down the known values next to their labels so you’re not hunting through the problem statement later.

Step 2: Set Up the Law of Cosines Formula

Now that you know what you’re solving for, set up the equation.

The standard form is:

c² = a² + b² – 2ab cos(C)

But the formula adapts depending on which side or angle you’re solving for.

The key is to make sure the side you’re solving for matches the angle in the cosine term.

If you’re solving for side c, the formula is c² = a² + b² – 2ab cos(C).

If you’re solving for side a, the formula is a² = b² + c² – 2bc cos(A).

If you’re solving for side b, the formula is b² = a² + c² – 2ac cos(B).

Substitute the known values into the equation.

If you’re solving for a side, plug in the two known sides and the angle between them.

If you’re solving for an angle, plug in all three sides and isolate the cosine term.

Step 3: Solve for the Unknown

Now you just finish the calculation.

If you’re solving for a side, square the two known sides, subtract the correction term, and take the square root of the result.

If you’re solving for an angle, rearrange the equation to isolate the cosine term, then take the inverse cosine to find the angle.

For example, if you’re solving for angle C and you’ve rearranged the equation to cos(C) = (a² + b² – c²) / (2ab), plug in your known values, calculate the fraction, then hit the inverse cosine button on your calculator to get the angle in degrees.

Make sure your calculator is in degree mode unless the problem specifies radians.

Double-check the units before you compare your answer to the choices.

With that laid out, let’s put these steps into practice.

Example Problem: Law of Cosines

The workflow handles any oblique triangle problem where you’re given enough information to solve for a missing side or angle.

The goal is to identify the configuration, set up the equation correctly, and execute the calculation without sign errors or mismatched labels.

Right now, you’re going to work through a real FE-style problem so the workflow becomes automatic.

The goal is clean structure—speed comes after a few reps.

With that laid out, let’s put these steps into practice.

This problem states:

Solution: Law of Cosines

When you see sides of 45 meters and 62 meters with a 73-degree angle between them, the fork isn’t whether you can solve it—it’s whether the formula is c² = a² + b² – 2ab cos(C) or if that minus sign becomes a plus for obtuse angles.

And then there’s the question: does it matter which side you call a versus b, or will swapping them break the calculation?

This is exactly why we use a workflow—it removes both questions.

The minus sign never changes, and the side labels only matter for matching the opposite angle.

The workflow turns that uncertainty into a repeatable three-step process you can trust under pressure.

Let’s walk it out step by step, the same way you would across the table.

Step 1: Identify What You’re Given and Label the Triangle

The first thing we need to do is read through the problem statement and identify what we’re working with.

We know two sides—45 meters and 62 meters—and we know the angle between them is 73 degrees.

We need to find the third side.

Let’s label the triangle.

We’ll call the two known sides a = 45 meters and b = 62 meters, and we’ll call the angle between them C = 73 degrees.

The side we’re solving for is c, which is opposite angle C.

This is a side-angle-side (SAS) configuration, which means we’re using the Law of Cosines to solve for the third side.

Step 2: Set Up the Law of Cosines Formula

Next, we need to set up the equation.

Since we’re solving for side c, the formula is:

c² = a² + b² – 2ab cos(C)

Now we substitute the known values:

c² = (45)² + (62)² – 2(45)(62) cos(73°)

Before we move on, let’s confirm the setup: side c is on the left, and angle C is in the cosine term.

They match because c is opposite C.

The setup is clean.

Step 3: Solve for the Unknown

Now we just finish it.

Let’s calculate each term:

(45)² = 2025

(62)² = 3844

2(45)(62) = 5580

cos(73°) ≈ 0.2924

Bringing those together, we get:

c² = 2025 + 3844 – 5580(0.2924)

c² = 5869 – 1631.59

c² = 4237.41

Taking the square root:

c ≈ 65.1 meters

So the final answer to this problem is B) 64 meters.

This tells us the third side of the triangular plot is approximately 64 meters, which makes sense given that it’s longer than the 45-meter side but shorter than the 62-meter side, and the 73-degree angle opens the triangle enough that the Pythagorean theorem would have underestimated the length.

Common Mistakes to Avoid on Law of Cosines Problems

Law of Cosines problems break at the exact moment you write c² = a² + b² + 2ab cos(C) instead of c² = a² + b² – 2ab cos(C).

You’ve labeled the triangle correctly, you’ve matched side c to angle C, you’re calculating with 45 and 62 and 73 degrees exactly as given—and then that plus sign adds 1,632 to your total instead of subtracting it, and you get c² = 7,501 instead of c² = 4,237.

Your final answer becomes 86.6 meters instead of 65.1 meters, and none of the answer choices are even close.

Mistake 1: Flipping the Minus Sign to a Plus

You write the formula as c² = a² + b² + 2ab cos(C) instead of c² = a² + b² – 2ab cos(C).

You’re moving fast, and somewhere in your memory you think the sign might change depending on whether the angle is acute or obtuse.

On the FE with our example, this gives you c² = 2025 + 3844 + 1632 = 7501, which leads to c ≈ 86.6 meters.

That’s more than 20 meters off the correct answer of 65.1 meters.

The plus sign adds length instead of adjusting for the oblique angle.

Your calculation is arithmetically perfect, but the wrong sign breaks the entire result.

Mistake 2: Mismatching Sides and Angles in the Formula

You set up c² = a² + b² – 2ab cos(C), but you pair sides a = 45 and b = 62 with angle B instead of angle C.

Or you’re solving for side a but you put angle C in the formula instead of angle A.

The side you’re solving for must be opposite the angle in the cosine term.

When you mismatch them, the correction term adjusts the wrong relationship.

Your answer comes out wrong by a factor large enough that none of the choices match, and you waste time recalculating the same broken setup.

Mistake 3: Forgetting to Take the Square Root

You calculate c² = 2025 + 3844 – 1632 = 4237, see that result, and write down 4237 meters as your final answer without taking the square root.

The formula gives you c², not c.

In our example, c² = 4237.41, which means c = √4237.41 ≈ 65.1 meters.

If you skip the square root, your answer is off by a factor of about 65, and it’s larger than the sum of the other two sides, which violates the triangle inequality.

Mistake 4: Using the Wrong Calculator Mode

You’re solving for an angle, you rearrange to get cos(C) = 0.4667, you hit inverse cosine, and your calculator returns 1.09.

None of the answer choices are close to 1.09, so you assume you set up the equation wrong.

What actually happened: your calculator was in radian mode, and 1.09 radians equals about 62.2 degrees.

The FE expects degrees unless specified otherwise.

You calculated correctly but reported the answer in the wrong units, which makes it look like your setup failed.

Mistake 5: Using Law of Cosines on the Wrong Configuration

You see a triangle problem, grab the Law of Cosines formula, and start plugging in numbers.

Halfway through, you realize the problem gave you two angles and one side, which isn’t a Law of Cosines setup.

Law of Cosines requires either two sides and the included angle (SAS), or all three sides (SSS).

If you have two angles and a side, or two sides and a non-included angle, you’re in a different configuration.

You’ve wasted 60-90 seconds on the wrong approach, and now you have to restart with a different tool.

Rules of Thumb for Law of Cosines Problems on the FE

You can label a triangle now, match side c to angle C, and write c² = a² + b² – 2ab cos(C) with the minus sign in the right place.

These rules are what keep you from writing 4237 meters when you calculated c² = 4237 and forgot the square root, or getting 1.09 when you needed 62.2 degrees because your calculator was in radian mode.

• The side and angle must match: The side you’re solving for is always opposite the angle in the cosine term. If you’re solving for side c, angle C is in the formula. If you’re solving for angle A, side a is in the formula. Before you substitute any numbers, verify the match. In our example with sides 45, 62 and angle 73°, we were solving for side c, so angle C = 73° went in the formula. If they don’t align, you’ve mislabeled the triangle and need to fix it before you calculate anything.
• The minus sign is not optional: The Law of Cosines formula is c² = a² + b² – 2ab cos(C). The correction term is always subtracted, not added. When we calculated 2025 + 3844 – 1632, that minus sign was what gave us 4237 instead of 7501. If you flip the sign, you’re adding length instead of adjusting for the oblique angle, and your answer will be too large. Write the formula down every time so you don’t second-guess the sign under pressure.
• Check your calculator mode before you start: The FE expects angles in degrees unless the problem specifies radians. If your calculator is in radian mode, your inverse cosine result will be numerically wrong in a way that doesn’t match any answer choice. Before you calculate any angle, glance at your calculator mode. If you’re expecting something between 30 and 90 degrees and you get 0.87, you’re in radians.
• Don’t skip the square root: When you’re solving for a side, the formula gives you c², not c. In our example, we got c² = 4237.41, which means c = √4237.41 ≈ 65.1 meters. If you write down 4237 meters without taking the square root, your answer is off by a factor of 65. Always check: did I solve for the side, or did I solve for the side squared?
• Sanity-check the magnitude: If you’re solving for a side and your answer is longer than the sum of the other two sides, something broke. In our example, c came out to 65.1 meters, which is less than 45 + 62 = 107 meters, so it passes. If you’re solving for an angle and your result is greater than 180 degrees, the setup is wrong. Use magnitude checks to catch errors before you commit to an answer.
• SAS and SSS are your triggers: Law of Cosines works in two configurations: when you know two sides and the angle between them (like our 45m, 62m, 73° example), or when you know all three sides. If the problem gives you two angles and a side, or two sides and an angle that isn’t between them, Law of Cosines isn’t the right tool. Identify the configuration before you start so you don’t waste time on the wrong approach.

Final Thoughts | Law of Cosines

Law of Cosines problems bleed time at the labeling step—not the calculation—and most students don’t see it coming.

They think the hard part is remembering whether it’s c² = a² + b² – 2ab cos(C) or if the sign flips.

So they write the formula, start calculating, and then realize 30 seconds later that they matched side a with angle B instead of angle A, and now the correction term is adjusting the wrong relationship.

That mismatch forces a restart.

You recalculate everything, verify your arithmetic twice, and burn 90 seconds fixing something that should have taken 10 seconds to prevent.

The workflow removes that time bleed by forcing you to label the triangle first and verify the side-angle match before you touch a single number.

Write “a opposite A, b opposite B, c opposite C” on your scratch work, confirm which side you’re solving for, then substitute.

That 10-second check saves you from a 90-second recalculation spiral.

The FE doesn’t care if you understand what the Law of Cosines means conceptually.

It tests whether you can set up c² = a² + b² – 2ab cos(C) with the right side opposite the right angle, execute the arithmetic without flipping the minus sign, and take the square root before comparing to answer choices—all while the clock runs and three other problems are waiting.

And that’s a structure problem, not a math problem.

Ready to keep building? Explore our complete FE Exam problem library here.

The gap between knowing the Law of Cosines exists and executing it confidently under exam pressure isn’t more practice problems—you can find those anywhere.

It’s knowing you’re studying the right topics, in the right order, with verification that you’re actually improving and not just repeating the same mistakes in different problems.

That uncertainty—not knowing if the hours you’re investing are actually building competence or just burning time—is what turns prep into a second-guessing spiral.

Prepineer closes that gap. You get a personalized roadmap that targets what the FE actually tests, structured practice that builds real confidence (not just familiarity), and coaching support when you’re stuck—so you walk into exam day knowing you’re ready, not hoping you studied enough. Start your free 7-day trial/ and see what focused prep with real structure feels like.