Generalized Pythagorean.
Reduces to Pythagoras when the included angle is 90 degrees, so it covers right and non-right cases.
Concept
AI Solver by AskSia
Law of Cosines Solver: applied step by step
Type or photograph the three sides or the SAS data of a triangle. AskSia applies c^2 = a^2 + b^2 - 2ab*cos(C) (or its rearrangement) to find a missing side or angle, with every arithmetic step shown.
Works with word problems, equations, code, and science prompts.
∫ 3x² · sin(x) dx
SubjectsCalculusAlgebraPhysicsChemistryBiologyCSStatisticsEcon
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asksia.ai/solver
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Problem
Find the area enclosed between y = x² and y = 2x in the first quadrant.
1
Find the points of intersection
Set the curves equal: x² = 2x → x(x - 2) = 0, so x = 0 and x = 2.
2
Identify which curve is on top
On (0, 2), 2x > x², so y = 2x is the upper curve.
3
Set up the area integral
A = ∫₀² (2x - x²) dx
4
Integrate term-by-term
A = [ x² - x³/3 ]₀²
Final Answer
A = 4/3 sq. units
Quick Answer
What is the law of cosines?
The law of cosines is a generalization of the Pythagorean theorem to any triangle (not just right triangles). It states c^2 = a^2 + b^2 - 2ab*cos(C), where a, b, c are the side lengths and C is the angle opposite side c. When C = 90 degrees, cos(C) = 0 and the formula reduces to the Pythagorean theorem. Use the law of cosines to find a third side given two sides and the included angle (SAS) or to find any angle given all three sides (SSS).
98%
solution accuracy
50M+
problems solved
~1.5s
avg solve time
A+
study-ready explanations
Why AskSia Solver
Why students use AskSia for Law of Cosines.
Every step transparent, every answer self-checked.
SSS angle finding.
Three sides known? Rearrange to cos(C) = (a^2 + b^2 - c^2)/(2ab), then arccos.
SSS
SAS side finding.
Two sides and included angle? Direct formula gives the third side.
SAS
Sign of cosine.
cos(C) > 0 for acute angle, cos(C) < 0 for obtuse. The formula handles both correctly.
Sign-aware
Photo, paste, or type.
Snap handwritten or printed problems with your phone, paste from any online homework portal, or type with full LaTeX support.
Multi-modal input
Verified by AskSia.
Every answer gets a self-check pass. Sia catches sign errors and algebra mistakes before you submit your homework.
Self-checked
How It Works
Solve any Law of Cosines problem in three steps.
Step 01
Enter the problem.
Type the expression, paste from your homework, snap a photo, or speak it. AskSia parses your input and identifies the structure.
Input mode
Snap a Photo
Textbook, handwriting, screenshot
Paste Text
Word problem or equation
Calculator
LaTeX-ready equation editor
Step 02
AskSia picks the method.
Based on the problem structure, AskSia chooses the cleanest solution path and labels each step with the operation performed.
1
Set curves equal
x² = 2x → x = 0, x = 2
2
Set up the integral
A = ∫₀² (2x - x²) dx
3
Evaluate
A = [x² - x³/3]₀² = 4/3
Step 03
Read the verified answer.
Final result appears with a substitution or composition check. Practice problems on the same concept are one tap away.
Auto-generated diagram
Region between y = 2x and y = x² — area = 4/3
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Every solve syncs across Web, iOS, and Android — start it at your desk, finish on your phone.
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Full study studio
Split-panel interface with the worked solution on the left, the auto-generated diagram and AI tutor chat on the right.
Drag & drop image upload + LaTeX equation editor
Auto-generated diagrams render alongside steps
Side-panel AI tutor chat for hints and alt methods
Export to PDF, DOCX, Notion, or Google Docs
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Calculus
98% verified
1.4s
Step 4 of 4 · Evaluate
A = [x² - x³/3]₀² = 4/3
Mobile App
Snap & solve, anywhere
Open the camera, frame the problem, and the worked solution plus diagram appear in seconds.
One-tap snap-and-solve on iOS and Android
Pinch-to-zoom diagrams, swipe between steps
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1.4s
Area between y=2x & y=x²
A = 4/3 sq. units ✓
Use Cases
What the Law of Cosines solver covers.
Find third side (SAS).
Two sides and included angle given. Direct application.
SAS
Find an angle (SSS).
Three sides given. Solve cos(angle) = ratio and take arccos.
SSS
Triangle with obtuse angle.
cos becomes negative. AskSia handles correctly.
Obtuse
Verify Pythagorean.
For C = 90 degrees, law of cosines reduces to Pythagorean theorem.
Check
Distance applications.
Find distance across a lake, between two points with known angle.
Applied
Verify your homework.
Paste your candidate answer and the original problem. AskSia walks the work, flags any divergent step, and tells you the correct final value.
Answer check
Compare
AskSia vs. ChatGPT,
Photomath & Symbolab.
General chatbots hallucinate. Photo solvers stop at math. AskSia is built for actual coursework with verified accuracy, visual learning, and every subject.
| Feature | AskSia Solver | ChatGPT | Photo Solvers |
|---|---|---|---|
| Solution accuracy | ✓ 98% | ~70-85%, hallucinations | ~90%, math only |
| Auto-generated diagrams | ✓ Every solve | Inconsistent / broken | Graphs only, math-only |
| Step-by-step explanations | ✓ Numbered + plain English | Inconsistent depth | ✓ Math steps |
| Subject coverage | ✓ Math, Physics, Chem, Bio, CS, Econ | ✓ Wide but unverified | Math only |
| Photo input | ✓ Handwriting + diagrams + code | Photos OK, weak on handwriting | ✓ Math photos only |
| Answer verification | ✓ Self-checked before display | No verification | Math engine only |
| Tutor follow-ups | ✓ Hints, alt methods, ELI5 | ✓ General chat | Not available |
| Practice and flashcards | ✓ One-tap from any solve | Manual prompting | Not available |
| Code debugging | ✓ Python, Java, C++, SQL... | ✓ Yes | Not available |
| Free to start | ✓ Daily solves, no card | Limited model access | Steps locked behind paywall |
FAQ
Frequently asked questions.
When do I use law of cosines versus law of sines?
Law of cosines is best when you have SSS (three sides) or SAS (two sides and included angle). Law of sines is best when you have an angle and its opposite side, plus another piece (ASA, AAS, or SSA). AskSia picks based on what you provide and explains the choice.
How do I solve for an angle using law of cosines?
Rearrange c^2 = a^2 + b^2 - 2ab*cos(C) to solve for cos(C): cos(C) = (a^2 + b^2 - c^2) / (2ab). Compute the right-hand side, then apply arccos to find the angle. AskSia shows the rearrangement, the substitution, and the arccos step explicitly.
Does the law of cosines work for obtuse triangles?
Yes. The only difference is that cos(C) becomes negative when C is between 90 and 180 degrees. This makes -2ab*cos(C) positive, increasing c^2 and giving a longer c than would occur in a right triangle. AskSia keeps signs straight and reports angles in the correct range.
How is the law of cosines proved?
By placing the triangle in a coordinate system: one vertex at the origin, one side along the positive x-axis, and the third vertex at (b*cos(A), b*sin(A)) for an angle A at the origin. The distance formula applied to the three vertices yields the law of cosines after algebraic simplification. AskSia can show this derivation on request.
How accurate is AskSia?
AskSia is engineered for accuracy on standard high school and college coursework. Accuracy comes from subject-specialized models, a symbolic verification pass that catches arithmetic errors, and a self-check step that re-derives the answer before showing it to you.
Can I get practice problems and flashcards?
Yes. After any solve, ask Sia to generate similar practice problems at SAT, ACT, AP, IB, or college difficulty, or build a flashcard set on the underlying concept in one tap. Useful for exam prep and spaced repetition before a quiz, midterm, or final.
How much does AskSia cost?
AskSia has a free plan that includes daily solves across all subjects. AskSia Pro and Super include unlimited solves, advanced subjects, the full AI tutor companion, exports, and priority response speed. See pricing for details.
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SSS or SAS, the third piece, every time.
Join 2M+ students using AskSia to solve law of cosines problems step-by-step. Photo input, plain-English explanations, and a verification check on every solve.
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