Law of Sines
Trigonometry, Geometry
Intro: Enter any valid combination containing at least one side and its opposite angle. We compute the remaining sides/angles, detect SSA ambiguity, and show unit-aware steps (° or radians).
Worked example
- Given B = 47°, C = 71°, and a = 8, find A, b, c.
- Angle sum: $$A = 180^{\circ} - (B + C) = 180^{\circ} - (47^{\circ} + 71^{\circ}) = 62^{\circ}.$$
- Law of Sines setup: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}.$$
- Solve for b: $$b = \frac{\sin B}{\sin A}\,a = \frac{\sin 47^{\circ}}{\sin 62^{\circ}}\cdot 8.$$
- Compute numerically (4 d.p.): $\sin 47^{\circ} \approx 0.7314$, $\sin 62^{\circ} \approx 0.8829$, so $$b \approx \frac{0.7314}{0.8829}\cdot 8 \approx 6.625.$$
- Solve for c: $$c = \frac{\sin C}{\sin A}\,a = \frac{\sin 71^{\circ}}{\sin 62^{\circ}}\cdot 8.$$
- $\sin 71^{\circ} \approx 0.9455$ → $$c \approx \frac{0.9455}{0.8829}\cdot 8 \approx 8.561.$$
- Summary: $$\boxed{A=62^{\circ}},\; \boxed{b\approx 6.625},\; \boxed{c\approx 8.561}.$$
- Check (optional): ratios $a/\sin A \approx 8/\sin 62^{\circ} \approx 9.061$; $b/\sin B \approx 6.625/\sin 47^{\circ} \approx 9.062$; $c/\sin C \approx 8.561/\sin 71^{\circ} \approx 9.057$ (agreement within rounding).
- Given a = 7, b = 10, and A = 30°, find possible B, C, c.
- Law of Sines to find $\sin B$: $$\frac{\sin B}{b} = \frac{\sin A}{a} \;\Rightarrow\; \sin B = b\,\frac{\sin A}{a} = 10\,\frac{\sin 30^{\circ}}{7} = 10\cdot\frac{0.5}{7} = \frac{5}{7} \approx 0.7143.$$
- Primary angle: $$B_1 = \arcsin(5/7) \approx 45.58^{\circ}.$$
- Ambiguity: since $\sin(\theta) = \sin(180^{\circ}-\theta)$, a second angle may exist: $$B_2 = 180^{\circ} - B_1 \approx 134.42^{\circ}.$$
- Check triangle viability with $A=30^{\circ}$:
- • Case 1: $B_1\approx 45.58^{\circ}$ → $$C_1 = 180^{\circ} - A - B_1 \approx 180^{\circ} - 30^{\circ} - 45.58^{\circ} = 104.42^{\circ}.$$
- • Case 2: $B_2\approx 134.42^{\circ}$ → $$C_2 = 180^{\circ} - 30^{\circ} - 134.42^{\circ} = 15.58^{\circ}.$$ Both are positive → two valid triangles.
- Find side c via Law of Sines:
- • Case 1: $$c_1 = \frac{\sin C_1}{\sin A}\,a = \frac{\sin 104.42^{\circ}}{\sin 30^{\circ}}\cdot 7 \approx \frac{0.9700}{0.5}\cdot 7 \approx 13.58.$$
- • Case 2: $$c_2 = \frac{\sin C_2}{\sin A}\,a = \frac{\sin 15.58^{\circ}}{\sin 30^{\circ}}\cdot 7 \approx \frac{0.2691}{0.5}\cdot 7 \approx 3.767.$$
- Conclusions:
- • Triangle 1: $$\boxed{B\approx 45.58^{\circ}},\; \boxed{C\approx 104.42^{\circ}},\; \boxed{c\approx 13.58}.$$
- • Triangle 2: $$\boxed{B\approx 134.42^{\circ}},\; \boxed{C\approx 15.58^{\circ}},\; \boxed{c\approx 3.77}.$$
- Notes: If $b\sin A / a > 1$ → **no solution**; if $=1$ → **one right triangle**; if $<1$ → **0/1/2** solutions depending on geometry.
FAQs
Degrees or radians?
We accept both. Include the ° symbol for degrees; otherwise values are treated as radians.
When is Law of Sines best?
Use it for ASA or AAS directly. For SAS or SSS, start with Law of Cosines, then switch to Law of Sines.
SSA ambiguity?
SSA may yield 0, 1, or 2 triangles. We compute the principal angle from arcsin and test the supplementary angle.
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,\lnworks 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.