Law of Sines: Solving a Triangle

The Law of Sines says that in any triangle, each side is proportional to the sine of its opposite angle:

$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$

This single ratio (which equals the diameter of the circumscribed circle) lets you solve for unknown sides given an angle and its opposite side, plus another angle.

Step-by-Step Solution

Given: $A = 40°$, $B = 60°$, $a = 10$. Find: $b$, $c$, and $C$.

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Step 1 — Find angle $C$ using the angle-sum law.

The interior angles of any triangle sum to $180°$:

$A + B + C = 180°$

$C = 180° - A - B = 180° - 40° - 60° = 80°$

Step 2 — Apply the Law of Sines to find $b$.

$\dfrac{a}{\sin A} = \dfrac{b}{\sin B}$

Solve for $b$:

$b = a \cdot \dfrac{\sin B}{\sin A} = 10 \cdot \dfrac{\sin 60°}{\sin 40°}$

Compute: $\sin 60° \approx 0.8660$, $\sin 40° \approx 0.6428$.

$b = 10 \cdot \dfrac{0.8660}{0.6428} \approx 10 \cdot 1.3473 \approx 13.473$

Step 3 — Apply the Law of Sines to find $c$.

$c = a \cdot \dfrac{\sin C}{\sin A} = 10 \cdot \dfrac{\sin 80°}{\sin 40°}$

$\sin 80° \approx 0.9848$.

$c = 10 \cdot \dfrac{0.9848}{0.6428} \approx 10 \cdot 1.5320 \approx 15.320$

Step 4 — Verify the proportions hold.

Check that all three ratios are equal:

$\dfrac{a}{\sin A} = \dfrac{10}{0.6428} \approx 15.557$

$\dfrac{b}{\sin B} = \dfrac{13.473}{0.8660} \approx 15.558$

$\dfrac{c}{\sin C} = \dfrac{15.320}{0.9848} \approx 15.557$

All three within rounding error. ✓ (The common value, $\approx 15.557$, is the diameter of the circumscribed circle — a beautiful geometric fact.)

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Answer:

$\boxed{\;C = 80°,\quad b \approx 13.47,\quad c \approx 15.32\;}$

The triangle has angles $40°,\, 60°,\, 80°$ and sides $10,\, 13.47,\, 15.32$ — the sides line up in the same order as their opposite angles (smallest angle ↔ smallest side).

Try It

• Adjust angles A and B with the sliders. The visualization updates the triangle in real time and recomputes $C$, $b$, $c$.
• The HUD shows all three Law-of-Sines ratios — they should always agree.
• Notice that the side opposite the largest angle is always the largest side (and vice versa).