Triangle Solver (ASA/SSS/SAS/SSA)

Intro: Enter any valid combination (2 angles + 1 side, or 2 sides + 1 angle). We’ll apply Law of Sines/Cosines with unit-aware trig.

Worked example

• Given $a=8$, $b=12$, included angle $C=37°$, find $c, A, B$.



---



• Goal: Use **SAS** (two sides with included angle). First find side $c$ by the Law of Cosines; then angles $A$ and $B$ by the Law of Sines.



• Law of Cosines: $$c^2=a^2+b^2-2ab\cos C.$$



• Substitute $a=8$, $b=12$, $C=37^{\circ}$: $$c^2=8^2+12^2-2\cdot8\cdot12\cos 37^{\circ}=64+144-192\cos 37^{\circ}.$$



• Numerics (to 4 d.p.): $\cos 37^{\circ}\approx 0.7986$. So $$c^2\approx 208-192(0.7986)\approx 208-153.338\approx 54.662.$$



• Take the square root: $$c\approx \sqrt{54.662}\approx 7.393.$$



• Law of Sines: $$\frac{\sin A}{a}=\frac{\sin C}{c} \;\Rightarrow\; \sin A=\frac{a\sin C}{c}.$$



• Compute with $\sin 37^{\circ}\approx 0.6018$: $$\sin A\approx \frac{8\cdot0.6018}{7.393}\approx 0.6512,$$ hence $$A\approx \arcsin(0.6512)\approx 40.63^{\circ}.$$



• Third angle: $$B=180^{\circ}-A-C\approx 180^{\circ}-40.63^{\circ}-37^{\circ}\approx 102.37^{\circ}.$$



• Summary (rounded): $$\boxed{c\approx 7.393,\quad A\approx 40.63^{\circ},\quad B\approx 102.37^{\circ}}.$$



• Checks (optional): (i) Angle sum $A+B+C\approx 180^{\circ}$ ✔️; (ii) Use Law of Sines to recover $b$: $\dfrac{\sin B}{b}\approx\dfrac{\sin C}{c}$ → both sides $\approx 0.108$ ✔️.

Why choose MathGPT?

• Get clear, step-by-step solutions that explain the “why,” not just the answer.
• See the rules used at each step (power, product, quotient, chain, and more).
• Optional animated walk-throughs to make tricky ideas click faster.
• Clean LaTeX rendering for notes, homework, and study guides.

How this calculator works

• Type or paste your function (LaTeX like \sin, \ln works too).
• Press Generate a practice question button to generate the derivative and the full reasoning.
• Review each step to understand which rule was applied and why.
• Practice with similar problems to lock in the method.

Related

• Law of Sines
• Law of Cosines