Laws of Sines and Cosines - Tenth Grade
Introduction
Purpose: Solve ANY triangle (not just right triangles)
Two Main Laws:
• Law of Sines: Relates sides to opposite angles using sine ratios
• Law of Cosines: Extension of Pythagorean theorem for any triangle
Applications: Navigation, surveying, engineering, astronomy, physics
Key Difference from Right Triangle Trig: These work for ALL triangles, including oblique triangles
Two Main Laws:
• Law of Sines: Relates sides to opposite angles using sine ratios
• Law of Cosines: Extension of Pythagorean theorem for any triangle
Applications: Navigation, surveying, engineering, astronomy, physics
Key Difference from Right Triangle Trig: These work for ALL triangles, including oblique triangles
1. Law of Sines
Law of Sines (Sine Rule): In any triangle, the ratio of a side length to the sine of its opposite angle is constant
Also Known As: Sine Rule
Key Idea: Connects each side with its opposite angle
Works for: ALL triangles (acute, obtuse, right)
Also Known As: Sine Rule
Key Idea: Connects each side with its opposite angle
Works for: ALL triangles (acute, obtuse, right)
Law of Sines Formula:
For triangle ABC with sides a, b, c opposite to angles A, B, C:
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
Alternative Form (Reciprocal):
$$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$$
In Words: Each side divided by the sine of its opposite angle gives the same value
Extended Form (with circumradius R):
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$$
where R is the radius of the circumscribed circle
For triangle ABC with sides a, b, c opposite to angles A, B, C:
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
Alternative Form (Reciprocal):
$$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$$
In Words: Each side divided by the sine of its opposite angle gives the same value
Extended Form (with circumradius R):
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$$
where R is the radius of the circumscribed circle
When to Use Law of Sines
Use Law of Sines when you have:
Case 1: AAS (Angle-Angle-Side)
• Two angles and one side
• Example: A = 50°, B = 60°, a = 8
Case 2: ASA (Angle-Side-Angle)
• Two angles and the included side
• Example: A = 40°, C = 70°, b = 12
Case 3: SSA (Side-Side-Angle) - AMBIGUOUS CASE
• Two sides and an angle opposite one of them
• May have 0, 1, or 2 solutions!
• Example: a = 10, b = 12, A = 30°
Case 1: AAS (Angle-Angle-Side)
• Two angles and one side
• Example: A = 50°, B = 60°, a = 8
Case 2: ASA (Angle-Side-Angle)
• Two angles and the included side
• Example: A = 40°, C = 70°, b = 12
Case 3: SSA (Side-Side-Angle) - AMBIGUOUS CASE
• Two sides and an angle opposite one of them
• May have 0, 1, or 2 solutions!
• Example: a = 10, b = 12, A = 30°
Example 1: AAS Case
Given: A = 40°, B = 60°, a = 10. Find side b.
Step 1: Set up Law of Sines:
$$\frac{a}{\sin A} = \frac{b}{\sin B}$$
Step 2: Substitute values:
$$\frac{10}{\sin 40°} = \frac{b}{\sin 60°}$$
Step 3: Solve for b:
$$\frac{10}{0.6428} = \frac{b}{0.8660}$$
$$15.56 = \frac{b}{0.8660}$$
$$b = 15.56 \times 0.8660 = 13.47$$
Answer: b ≈ 13.47
Given: A = 40°, B = 60°, a = 10. Find side b.
Step 1: Set up Law of Sines:
$$\frac{a}{\sin A} = \frac{b}{\sin B}$$
Step 2: Substitute values:
$$\frac{10}{\sin 40°} = \frac{b}{\sin 60°}$$
Step 3: Solve for b:
$$\frac{10}{0.6428} = \frac{b}{0.8660}$$
$$15.56 = \frac{b}{0.8660}$$
$$b = 15.56 \times 0.8660 = 13.47$$
Answer: b ≈ 13.47
Example 2: Find an angle
Given: a = 8, b = 10, A = 35°. Find angle B.
Step 1: Set up Law of Sines:
$$\frac{a}{\sin A} = \frac{b}{\sin B}$$
Step 2: Substitute:
$$\frac{8}{\sin 35°} = \frac{10}{\sin B}$$
Step 3: Solve for sin B:
$$\frac{8}{0.5736} = \frac{10}{\sin B}$$
$$13.95 \times \sin B = 10$$
$$\sin B = \frac{10}{13.95} = 0.717$$
Step 4: Find B:
$$B = \sin^{-1}(0.717) \approx 45.8°$$
Answer: B ≈ 45.8°
Given: a = 8, b = 10, A = 35°. Find angle B.
Step 1: Set up Law of Sines:
$$\frac{a}{\sin A} = \frac{b}{\sin B}$$
Step 2: Substitute:
$$\frac{8}{\sin 35°} = \frac{10}{\sin B}$$
Step 3: Solve for sin B:
$$\frac{8}{0.5736} = \frac{10}{\sin B}$$
$$13.95 \times \sin B = 10$$
$$\sin B = \frac{10}{13.95} = 0.717$$
Step 4: Find B:
$$B = \sin^{-1}(0.717) \approx 45.8°$$
Answer: B ≈ 45.8°
The Ambiguous Case (SSA)
Ambiguous Case Warning!
When given SSA (two sides and angle opposite one side), you may have:
0 Solutions: No triangle exists (side too short)
1 Solution: One unique triangle
2 Solutions: Two different triangles possible!
Check: After finding sin B, if sin B > 1, no triangle exists
If sin B ≤ 1, there may be two angles: B and (180° - B)
When given SSA (two sides and angle opposite one side), you may have:
0 Solutions: No triangle exists (side too short)
1 Solution: One unique triangle
2 Solutions: Two different triangles possible!
Check: After finding sin B, if sin B > 1, no triangle exists
If sin B ≤ 1, there may be two angles: B and (180° - B)
2. Law of Cosines
Law of Cosines (Cosine Rule): Relates all three sides of a triangle to one angle
Also Known As: Cosine Rule
Extension of: Pythagorean theorem (when angle = 90°, it becomes a² + b² = c²)
Works for: ALL triangles
Also Known As: Cosine Rule
Extension of: Pythagorean theorem (when angle = 90°, it becomes a² + b² = c²)
Works for: ALL triangles
Law of Cosines Formulas:
For triangle ABC with sides a, b, c opposite to angles A, B, C:
Form 1 (to find side c):
$$c^2 = a^2 + b^2 - 2ab \cos C$$
Form 2 (to find side a):
$$a^2 = b^2 + c^2 - 2bc \cos A$$
Form 3 (to find side b):
$$b^2 = a^2 + c^2 - 2ac \cos B$$
To find an angle (rearranged):
$$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$
$$\cos B = \frac{a^2 + c^2 - b^2}{2ac}$$
$$\cos C = \frac{a^2 + b^2 - c^2}{2ab}$$
For triangle ABC with sides a, b, c opposite to angles A, B, C:
Form 1 (to find side c):
$$c^2 = a^2 + b^2 - 2ab \cos C$$
Form 2 (to find side a):
$$a^2 = b^2 + c^2 - 2bc \cos A$$
Form 3 (to find side b):
$$b^2 = a^2 + c^2 - 2ac \cos B$$
To find an angle (rearranged):
$$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$
$$\cos B = \frac{a^2 + c^2 - b^2}{2ac}$$
$$\cos C = \frac{a^2 + b^2 - c^2}{2ab}$$
When to Use Law of Cosines
Use Law of Cosines when you have:
Case 1: SAS (Side-Angle-Side)
• Two sides and the included angle
• Use to find the third side
• Example: a = 8, b = 10, C = 60°
Case 2: SSS (Side-Side-Side)
• All three sides
• Use to find any angle
• Example: a = 5, b = 7, c = 8
Case 1: SAS (Side-Angle-Side)
• Two sides and the included angle
• Use to find the third side
• Example: a = 8, b = 10, C = 60°
Case 2: SSS (Side-Side-Side)
• All three sides
• Use to find any angle
• Example: a = 5, b = 7, c = 8
Example 1: SAS - Find a side
Given: a = 12, b = 15, C = 60°. Find side c.
Step 1: Use Law of Cosines:
$$c^2 = a^2 + b^2 - 2ab \cos C$$
Step 2: Substitute values:
$$c^2 = 12^2 + 15^2 - 2(12)(15) \cos 60°$$
$$c^2 = 144 + 225 - 360(0.5)$$
$$c^2 = 369 - 180$$
$$c^2 = 189$$
Step 3: Take square root:
$$c = \sqrt{189} \approx 13.75$$
Answer: c ≈ 13.75
Given: a = 12, b = 15, C = 60°. Find side c.
Step 1: Use Law of Cosines:
$$c^2 = a^2 + b^2 - 2ab \cos C$$
Step 2: Substitute values:
$$c^2 = 12^2 + 15^2 - 2(12)(15) \cos 60°$$
$$c^2 = 144 + 225 - 360(0.5)$$
$$c^2 = 369 - 180$$
$$c^2 = 189$$
Step 3: Take square root:
$$c = \sqrt{189} \approx 13.75$$
Answer: c ≈ 13.75
Example 2: SSS - Find an angle
Given: a = 5, b = 7, c = 8. Find angle A.
Step 1: Use rearranged Law of Cosines:
$$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$
Step 2: Substitute:
$$\cos A = \frac{7^2 + 8^2 - 5^2}{2(7)(8)}$$
$$\cos A = \frac{49 + 64 - 25}{112}$$
$$\cos A = \frac{88}{112} = 0.7857$$
Step 3: Find angle:
$$A = \cos^{-1}(0.7857) \approx 38.2°$$
Answer: A ≈ 38.2°
Given: a = 5, b = 7, c = 8. Find angle A.
Step 1: Use rearranged Law of Cosines:
$$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$
Step 2: Substitute:
$$\cos A = \frac{7^2 + 8^2 - 5^2}{2(7)(8)}$$
$$\cos A = \frac{49 + 64 - 25}{112}$$
$$\cos A = \frac{88}{112} = 0.7857$$
Step 3: Find angle:
$$A = \cos^{-1}(0.7857) \approx 38.2°$$
Answer: A ≈ 38.2°
3. Solve a Triangle
Solving a Triangle: Finding all unknown sides and angles
Given: Usually 3 pieces of information (including at least one side)
Find: The remaining 3 pieces
Tools: Law of Sines, Law of Cosines, Angle Sum (A + B + C = 180°)
Given: Usually 3 pieces of information (including at least one side)
Find: The remaining 3 pieces
Tools: Law of Sines, Law of Cosines, Angle Sum (A + B + C = 180°)
Strategy for Solving Triangles:
Step 1: Identify What You Have
• ASA or AAS → Use Law of Sines
• SAS or SSS → Use Law of Cosines first
• SSA → Law of Sines (watch for ambiguous case!)
Step 2: Find Third Angle (if needed)
Use: A + B + C = 180°
Step 3: Find Missing Sides/Angles
Use appropriate law
Step 4: Verify
Check that all angles sum to 180°
Step 1: Identify What You Have
• ASA or AAS → Use Law of Sines
• SAS or SSS → Use Law of Cosines first
• SSA → Law of Sines (watch for ambiguous case!)
Step 2: Find Third Angle (if needed)
Use: A + B + C = 180°
Step 3: Find Missing Sides/Angles
Use appropriate law
Step 4: Verify
Check that all angles sum to 180°
Example 1: Solve triangle (AAS)
Given: A = 50°, B = 70°, a = 15. Find C, b, and c.
Step 1: Find angle C
$$C = 180° - 50° - 70° = 60°$$
Step 2: Find side b (Law of Sines)
$$\frac{a}{\sin A} = \frac{b}{\sin B}$$
$$\frac{15}{\sin 50°} = \frac{b}{\sin 70°}$$
$$b = \frac{15 \times \sin 70°}{\sin 50°} = \frac{15 \times 0.9397}{0.7660} = 18.4$$
Step 3: Find side c (Law of Sines)
$$\frac{a}{\sin A} = \frac{c}{\sin C}$$
$$c = \frac{15 \times \sin 60°}{\sin 50°} = \frac{15 \times 0.8660}{0.7660} = 16.95$$
Answer: C = 60°, b ≈ 18.4, c ≈ 16.95
Given: A = 50°, B = 70°, a = 15. Find C, b, and c.
Step 1: Find angle C
$$C = 180° - 50° - 70° = 60°$$
Step 2: Find side b (Law of Sines)
$$\frac{a}{\sin A} = \frac{b}{\sin B}$$
$$\frac{15}{\sin 50°} = \frac{b}{\sin 70°}$$
$$b = \frac{15 \times \sin 70°}{\sin 50°} = \frac{15 \times 0.9397}{0.7660} = 18.4$$
Step 3: Find side c (Law of Sines)
$$\frac{a}{\sin A} = \frac{c}{\sin C}$$
$$c = \frac{15 \times \sin 60°}{\sin 50°} = \frac{15 \times 0.8660}{0.7660} = 16.95$$
Answer: C = 60°, b ≈ 18.4, c ≈ 16.95
Example 2: Solve triangle (SAS)
Given: a = 10, b = 14, C = 45°. Find c, A, and B.
Step 1: Find side c (Law of Cosines)
$$c^2 = 10^2 + 14^2 - 2(10)(14) \cos 45°$$
$$c^2 = 100 + 196 - 280(0.7071)$$
$$c^2 = 296 - 198 = 98$$
$$c = \sqrt{98} \approx 9.90$$
Step 2: Find angle A (Law of Sines or Cosines)
Using Law of Sines:
$$\frac{a}{\sin A} = \frac{c}{\sin C}$$
$$\sin A = \frac{a \sin C}{c} = \frac{10 \times 0.7071}{9.90} = 0.714$$
$$A = \sin^{-1}(0.714) \approx 45.6°$$
Step 3: Find angle B
$$B = 180° - 45° - 45.6° = 89.4°$$
Answer: c ≈ 9.90, A ≈ 45.6°, B ≈ 89.4°
Given: a = 10, b = 14, C = 45°. Find c, A, and B.
Step 1: Find side c (Law of Cosines)
$$c^2 = 10^2 + 14^2 - 2(10)(14) \cos 45°$$
$$c^2 = 100 + 196 - 280(0.7071)$$
$$c^2 = 296 - 198 = 98$$
$$c = \sqrt{98} \approx 9.90$$
Step 2: Find angle A (Law of Sines or Cosines)
Using Law of Sines:
$$\frac{a}{\sin A} = \frac{c}{\sin C}$$
$$\sin A = \frac{a \sin C}{c} = \frac{10 \times 0.7071}{9.90} = 0.714$$
$$A = \sin^{-1}(0.714) \approx 45.6°$$
Step 3: Find angle B
$$B = 180° - 45° - 45.6° = 89.4°$$
Answer: c ≈ 9.90, A ≈ 45.6°, B ≈ 89.4°
4. Area of a Triangle: Sine Formula
Area Using Sine: Calculate triangle area when you know two sides and the included angle
Advantage: Don't need the height!
When to Use: Given SAS (two sides and angle between them)
Advantage: Don't need the height!
When to Use: Given SAS (two sides and angle between them)
Area Formula Using Sine:
For triangle with sides a, b and included angle C:
$$\text{Area} = \frac{1}{2}ab \sin C$$
Alternative Forms:
$$\text{Area} = \frac{1}{2}bc \sin A$$
$$\text{Area} = \frac{1}{2}ac \sin B$$
In Words: Area equals half the product of two sides times the sine of the included angle
Key Point: The angle MUST be between the two sides!
For triangle with sides a, b and included angle C:
$$\text{Area} = \frac{1}{2}ab \sin C$$
Alternative Forms:
$$\text{Area} = \frac{1}{2}bc \sin A$$
$$\text{Area} = \frac{1}{2}ac \sin B$$
In Words: Area equals half the product of two sides times the sine of the included angle
Key Point: The angle MUST be between the two sides!
Example 1: Find area with two sides and angle
Triangle has sides a = 8, b = 10, and included angle C = 60°. Find the area.
Step 1: Use area formula:
$$\text{Area} = \frac{1}{2}ab \sin C$$
Step 2: Substitute:
$$\text{Area} = \frac{1}{2}(8)(10) \sin 60°$$
$$\text{Area} = \frac{1}{2}(80)(0.8660)$$
$$\text{Area} = 40 \times 0.8660$$
$$\text{Area} = 34.64$$
Answer: Area ≈ 34.64 square units
Triangle has sides a = 8, b = 10, and included angle C = 60°. Find the area.
Step 1: Use area formula:
$$\text{Area} = \frac{1}{2}ab \sin C$$
Step 2: Substitute:
$$\text{Area} = \frac{1}{2}(8)(10) \sin 60°$$
$$\text{Area} = \frac{1}{2}(80)(0.8660)$$
$$\text{Area} = 40 \times 0.8660$$
$$\text{Area} = 34.64$$
Answer: Area ≈ 34.64 square units
Example 2: Find angle given area
Triangle has sides 6 and 8, and area 20. Find the included angle.
Step 1: Use area formula:
$$20 = \frac{1}{2}(6)(8) \sin C$$
Step 2: Solve for sin C:
$$20 = 24 \sin C$$
$$\sin C = \frac{20}{24} = 0.8333$$
Step 3: Find angle:
$$C = \sin^{-1}(0.8333) \approx 56.4°$$
Answer: Included angle ≈ 56.4°
Triangle has sides 6 and 8, and area 20. Find the included angle.
Step 1: Use area formula:
$$20 = \frac{1}{2}(6)(8) \sin C$$
Step 2: Solve for sin C:
$$20 = 24 \sin C$$
$$\sin C = \frac{20}{24} = 0.8333$$
Step 3: Find angle:
$$C = \sin^{-1}(0.8333) \approx 56.4°$$
Answer: Included angle ≈ 56.4°
5. Area of a Triangle: Using Law of Sines
Area with Law of Sines: When you have angles and one side
Method: First use Law of Sines to find needed sides, then use area formula
Alternative: Use extended sine formula with angles and circumradius
Method: First use Law of Sines to find needed sides, then use area formula
Alternative: Use extended sine formula with angles and circumradius
Area Formula with Law of Sines:
Method 1: Find sides first, then use sine area formula
1. Use Law of Sines to find needed sides
2. Apply: $\text{Area} = \frac{1}{2}ab \sin C$
Method 2: Direct formula with circumradius
$$\text{Area} = \frac{abc}{4R}$$
where R is circumradius, and $R = \frac{a}{2\sin A}$
Method 3: Using all three sides (Heron's Formula)
$$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$$
where $s = \frac{a+b+c}{2}$ (semi-perimeter)
Method 1: Find sides first, then use sine area formula
1. Use Law of Sines to find needed sides
2. Apply: $\text{Area} = \frac{1}{2}ab \sin C$
Method 2: Direct formula with circumradius
$$\text{Area} = \frac{abc}{4R}$$
where R is circumradius, and $R = \frac{a}{2\sin A}$
Method 3: Using all three sides (Heron's Formula)
$$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$$
where $s = \frac{a+b+c}{2}$ (semi-perimeter)
Example: Area with angles and one side
Given: A = 40°, B = 60°, a = 10. Find the area.
Step 1: Find angle C:
$$C = 180° - 40° - 60° = 80°$$
Step 2: Find side b using Law of Sines:
$$\frac{a}{\sin A} = \frac{b}{\sin B}$$
$$b = \frac{a \sin B}{\sin A} = \frac{10 \times \sin 60°}{\sin 40°}$$
$$b = \frac{10 \times 0.8660}{0.6428} = 13.47$$
Step 3: Calculate area:
$$\text{Area} = \frac{1}{2}ab \sin C$$
$$\text{Area} = \frac{1}{2}(10)(13.47) \sin 80°$$
$$\text{Area} = \frac{1}{2}(134.7)(0.9848)$$
$$\text{Area} = 66.3$$
Answer: Area ≈ 66.3 square units
Given: A = 40°, B = 60°, a = 10. Find the area.
Step 1: Find angle C:
$$C = 180° - 40° - 60° = 80°$$
Step 2: Find side b using Law of Sines:
$$\frac{a}{\sin A} = \frac{b}{\sin B}$$
$$b = \frac{a \sin B}{\sin A} = \frac{10 \times \sin 60°}{\sin 40°}$$
$$b = \frac{10 \times 0.8660}{0.6428} = 13.47$$
Step 3: Calculate area:
$$\text{Area} = \frac{1}{2}ab \sin C$$
$$\text{Area} = \frac{1}{2}(10)(13.47) \sin 80°$$
$$\text{Area} = \frac{1}{2}(134.7)(0.9848)$$
$$\text{Area} = 66.3$$
Answer: Area ≈ 66.3 square units
When to Use Which Law
| Given Information | Abbreviation | Use Which Law | Find |
|---|---|---|---|
| Two angles, one side | AAS or ASA | Law of Sines | Other sides and third angle |
| Two sides, included angle | SAS | Law of Cosines | Third side first, then angles |
| All three sides | SSS | Law of Cosines | All angles |
| Two sides, non-included angle | SSA | Law of Sines | Other angle (ambiguous!) |
Formulas Quick Reference
| Law/Formula | Equation | Use |
|---|---|---|
| Law of Sines | $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ | AAS, ASA, SSA cases |
| Law of Cosines (for side) | $c^2 = a^2 + b^2 - 2ab\cos C$ | SAS case - find side |
| Law of Cosines (for angle) | $\cos A = \frac{b^2+c^2-a^2}{2bc}$ | SSS case - find angle |
| Area (Sine Formula) | $\text{Area} = \frac{1}{2}ab\sin C$ | Two sides + included angle |
| Heron's Formula | $A = \sqrt{s(s-a)(s-b)(s-c)}$ | Three sides known |
| Angle Sum | $A + B + C = 180°$ | Find third angle |
Problem-Solving Strategy Chart
| Step | Action | Example |
|---|---|---|
| 1. Identify | What information do you have? | AAS, SAS, SSS, ASA, or SSA? |
| 2. Choose Law | Law of Sines or Cosines? | SAS → Cosines; AAS → Sines |
| 3. Find Missing Angle | Use angle sum if needed | C = 180° - A - B |
| 4. Solve | Apply formula and calculate | Substitute and solve |
| 5. Verify | Check angles sum to 180° | Does A + B + C = 180°? |
Common Mistakes to Avoid
| Mistake | Correct Approach |
|---|---|
| Using Law of Sines for SAS | Use Law of Cosines for SAS (included angle) |
| Forgetting ambiguous case (SSA) | Check for two possible triangles when using SSA |
| Wrong angle in sine area formula | Angle MUST be between the two sides |
| Calculator in radians instead of degrees | Always check calculator mode (DEG not RAD) |
| Mixing up sides and angles | Side 'a' is opposite angle 'A', etc. |
Key Relationships Summary
| Concept | Formula/Rule | Note |
|---|---|---|
| Opposite pairs | Side a is opposite angle A | Labeling convention |
| Angle sum | A + B + C = 180° | Always true for triangles |
| Law of Sines ratio | All three ratios are equal | Can use any two at a time |
| Pythagorean special case | When C = 90°, $c^2 = a^2 + b^2$ | Law of Cosines reduces to this |
| Area with sine | Needs included angle | Half × product × sine |
Success Tips for Laws of Sines and Cosines:
✓ Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ (use for AAS, ASA, SSA)
✓ Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos C$ (use for SAS, SSS)
✓ Always check which law applies based on given information!
✓ SSA is ambiguous - may have 0, 1, or 2 solutions
✓ Area with sine: $A = \frac{1}{2}ab\sin C$ (angle must be included!)
✓ For SAS: use Law of Cosines first, then Law of Sines
✓ For SSS: use Law of Cosines to find angles
✓ Always verify: angles should sum to 180°
✓ Check calculator mode: DEGREE not RADIAN
✓ Label correctly: side 'a' opposite angle 'A'
✓ Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ (use for AAS, ASA, SSA)
✓ Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos C$ (use for SAS, SSS)
✓ Always check which law applies based on given information!
✓ SSA is ambiguous - may have 0, 1, or 2 solutions
✓ Area with sine: $A = \frac{1}{2}ab\sin C$ (angle must be included!)
✓ For SAS: use Law of Cosines first, then Law of Sines
✓ For SSS: use Law of Cosines to find angles
✓ Always verify: angles should sum to 180°
✓ Check calculator mode: DEGREE not RADIAN
✓ Label correctly: side 'a' opposite angle 'A'