📐 Laws of Sines & Cosines

Essential for solving non-right triangles. These laws extend trigonometry beyond right triangles and are frequently tested on AMC.

🎯 Key Ideas

Law of Sines: Relates sides and angles in any triangle: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Law of Cosines: Generalizes the Pythagorean theorem: $c^2 = a^2 + b^2 - 2ab\cos C$

Area Formula: $A = \frac{1}{2}ab\sin C$ (most useful area formula for AMC)

📐 Law of Sines

Formula

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

When to Use

  • AAS (Angle-Angle-Side): Two angles and one side
  • ASA (Angle-Side-Angle): Two angles and included side
  • SSA (Side-Side-Angle): Two sides and non-included angle (ambiguous case)

Example: Solve triangle with $A = 30°$, $B = 45°$, $a = 10$

Solution:

  1. Find third angle: $C = 180° - 30° - 45° = 105°$
  2. Apply Law of Sines: $\frac{10}{\sin 30°} = \frac{b}{\sin 45°} = \frac{c}{\sin 105°}$
  3. Solve for $b$: $b = \frac{10 \sin 45°}{\sin 30°} = \frac{10 \cdot \frac{\sqrt{2}}{2}}{\frac{1}{2}} = 10\sqrt{2}$
  4. Solve for $c$: $c = \frac{10 \sin 105°}{\sin 30°} = \frac{10 \sin 105°}{\frac{1}{2}} = 20\sin 105°$

📐 Law of Cosines

Formula

$$c^2 = a^2 + b^2 - 2ab\cos C$$

When to Use

  • SSS (Side-Side-Side): Three sides given
  • SAS (Side-Angle-Side): Two sides and included angle

Example: Solve triangle with $a = 5$, $b = 7$, $C = 60°$

Solution:

  1. Apply Law of Cosines: $c^2 = 5^2 + 7^2 - 2(5)(7)\cos 60°$
  2. Calculate: $c^2 = 25 + 49 - 70 \cdot \frac{1}{2} = 74 - 35 = 39$
  3. Solve: $c = \sqrt{39}$
  4. Find other angles using Law of Sines: $\frac{5}{\sin A} = \frac{\sqrt{39}}{\sin 60°}$

⚠️ Ambiguous Case (SSA)

When given two sides and a non-included angle, there may be:

  • No solution (side too short)
  • One solution (right triangle or side just long enough)
  • Two solutions (side long enough to form two different triangles)

Decision Process

For triangle with sides $a, b$ and angle $A$ opposite side $a$:

  1. Calculate height: $h = b\sin A$
  2. Compare $a$ with $h$:
    • If $a < h$: No solution
    • If $a = h$: One solution (right triangle)
    • If $h < a < b$: Two solutions
    • If $a \geq b$: One solution

Example: Solve triangle with $a = 8$, $b = 10$, $A = 30°$

Solution:

  1. Calculate height: $h = 10\sin 30° = 10 \cdot \frac{1}{2} = 5$
  2. Compare: $a = 8 > h = 5$ and $a = 8 < b = 10$, so two solutions
  3. Find angle $B$: $\sin B = \frac{b\sin A}{a} = \frac{10 \sin 30°}{8} = \frac{5}{8}$
  4. Two possible angles: $B_1 = \arcsin\frac{5}{8}$ and $B_2 = 180° - \arcsin\frac{5}{8}$
  5. Complete both triangles using Law of Sines

📊 Area Formulas

Standard Area Formula

$$A = \frac{1}{2}ab\sin C$$

Heron’s Formula (when all three sides known)

$$A = \sqrt{s(s-a)(s-b)(s-c)} \text{ where } s = \frac{a+b+c}{2}$$

Example: Find area of triangle with $a = 6$, $b = 8$, $C = 45°$

Solution:

  • $A = \frac{1}{2} \cdot 6 \cdot 8 \cdot \sin 45° = \frac{1}{2} \cdot 48 \cdot \frac{\sqrt{2}}{2} = 12\sqrt{2}$

🎯 AMC-Style Worked Example

Problem: In triangle $ABC$, $a = 7$, $b = 8$, and $c = 9$. Find the area and all angles.

Solution:

  1. Find area using Heron’s formula:

    • $s = \frac{7+8+9}{2} = 12$
    • $A = \sqrt{12(12-7)(12-8)(12-9)} = \sqrt{12 \cdot 5 \cdot 4 \cdot 3} = \sqrt{720} = 12\sqrt{5}$

  2. Find angles using Law of Cosines:

    • $\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{8^2 + 9^2 - 7^2}{2 \cdot 8 \cdot 9} = \frac{64 + 81 - 49}{144} = \frac{96}{144} = \frac{2}{3}$
    • $A = \arccos\frac{2}{3}$
    • Similarly: $\cos B = \frac{7^2 + 9^2 - 8^2}{2 \cdot 7 \cdot 9} = \frac{49 + 81 - 64}{126} = \frac{66}{126} = \frac{11}{21}$
    • $B = \arccos\frac{11}{21}$
    • $C = 180° - A - B$

Answer: Area = $12\sqrt{5}$, angles = $\arccos\frac{2}{3}$, $\arccos\frac{11}{21}$, and $180° - A - B$

🔍 Common Traps & Fixes

Trap: Forgetting the ambiguous case

Fix: Always check if SSA case has 0, 1, or 2 solutions before solving.

Trap: Wrong angle in Law of Cosines

Fix: The angle must be opposite the side you’re solving for: $c^2 = a^2 + b^2 - 2ab\cos C$.

Trap: Using wrong area formula

Fix: Use $A = \frac{1}{2}ab\sin C$ when you have two sides and included angle.

Trap: Angle sum not 180°

Fix: Always verify that $A + B + C = 180°$ in your final answer.

📋 Quick Reference

Law of Sines

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

Law of Cosines

$$c^2 = a^2 + b^2 - 2ab\cos C$$

Area Formulas

  • $A = \frac{1}{2}ab\sin C$ (two sides and included angle)
  • $A = \sqrt{s(s-a)(s-b)(s-c)}$ (three sides, $s = \frac{a+b+c}{2}$)

Ambiguous Case (SSA)

  • $a < b\sin A$: No solution
  • $a = b\sin A$: One solution (right triangle)
  • $b\sin A < a < b$: Two solutions
  • $a \geq b$: One solution

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