When To Use Law Of Cosines And Law Of Sines

When faced with solving triangles, the Law of Sines and the Law of Cosines are indispensable tools. Understanding when to apply each is crucial for accurate and efficient problem-solving. This article will delve into the nuances of both laws, providing a comprehensive guide on when to use each, supplemented with examples and practical insights.

Understanding the Law of Sines

The Law of Sines establishes a relationship between the angles of a triangle and the lengths of their opposite sides. Specifically, it states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides and angles in a triangle. Mathematically, it is expressed as:

a / sin(A) = b / sin(B) = c / sin(C)

Where:

a, b, and c are the lengths of the sides of the triangle.
A, B, and C are the angles opposite those sides, respectively.

When to Use the Law of Sines

The Law of Sines is particularly useful in two primary scenarios:

Angle-Side-Angle (ASA) or Angle-Angle-Side (AAS): When you know two angles and one side of a triangle, you can use the Law of Sines to find the remaining sides.
- ASA: You are given two angles and the included side (the side between the two angles).
- AAS: You are given two angles and a non-included side (a side not between the two angles).
Side-Side-Angle (SSA): This is also known as the ambiguous case because it can lead to one solution, two solutions, or no solution. You are given two sides and an angle opposite one of those sides.
- The SSA case requires careful analysis to determine the number of possible triangles that can be formed.

Example of Law of Sines (ASA)

Consider a triangle where angle A = 30°, angle B = 70°, and side c = 10 cm. Find the length of side a.

Step 1: Identify the knowns.
- Angle A = 30°
- Angle B = 70°
- Side c = 10 cm
Step 2: Use the Law of Sines.
```
a / sin(A) = c / sin(C)
```
Step 3: Find angle C. Since the sum of angles in a triangle is 180°,
```
C = 180° - A - B = 180° - 30° - 70° = 80°
```

Step 4: Substitute the values and solve for a.

a / sin(30°) = 10 / sin(80°)
a = (10 * sin(30°)) / sin(80°)
a ≈ (10 * 0.5) / 0.9848
a ≈ 5.077 cm

Thus, the length of side a is approximately 5.077 cm.

Example of Law of Sines (SSA) - The Ambiguous Case

Suppose you have a triangle with side a = 20 cm, side b = 15 cm, and angle A = 40°. Find angle B.

Step 1: Identify the knowns.
- Side a = 20 cm
- Side b = 15 cm
- Angle A = 40°
Step 2: Use the Law of Sines.
```
a / sin(A) = b / sin(B)
```

Step 3: Substitute the values and solve for sin(B).

20 / sin(40°) = 15 / sin(B)
sin(B) = (15 * sin(40°)) / 20
sin(B) ≈ (15 * 0.6428) / 20
sin(B) ≈ 0.4821

Step 4: Find angle B.
```
B = arcsin(0.4821)
B ≈ 28.82°
```

However, since sine values are positive in both the first and second quadrants, we need to check for a second possible angle:

B' = 180° - B
B' ≈ 180° - 28.82°
B' ≈ 151.18°

Now, check if this second angle is viable:

A + B' = 40° + 151.18° = 191.18°

Since the sum of A and B' is greater than 180°, B' is not a valid solution. Therefore, there is only one possible triangle in this case, with angle B ≈ 28.82°.

Understanding the Law of Cosines

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is particularly useful when you do not have an angle and its opposite side. The Law of Cosines comes in three forms, each isolating a different angle:

a² = b² + c² - 2bc * cos(A)
b² = a² + c² - 2ac * cos(B)
c² = a² + b² - 2ab * cos(C)

Where:

a, b, and c are the lengths of the sides of the triangle.
A, B, and C are the angles opposite those sides, respectively.

When to Use the Law of Cosines

The Law of Cosines is best applied in the following situations:

Side-Angle-Side (SAS): When you know two sides and the included angle (the angle between the two sides), you can use the Law of Cosines to find the third side.
Side-Side-Side (SSS): When you know all three sides of a triangle, you can use the Law of Cosines to find any of the angles.

Example of Law of Cosines (SAS)

Consider a triangle where side a = 5 cm, side b = 8 cm, and angle C = 77°. Find the length of side c.

Step 1: Identify the knowns.
- Side a = 5 cm
- Side b = 8 cm
- Angle C = 77°
Step 2: Use the Law of Cosines.
```
c² = a² + b² - 2ab * cos(C)
```

Step 3: Substitute the values and solve for c.

c² = 5² + 8² - 2 * 5 * 8 * cos(77°)
c² = 25 + 64 - 80 * cos(77°)
c² ≈ 89 - 80 * 0.2250
c² ≈ 89 - 18
c² ≈ 71
c ≈ √71
c ≈ 8.426 cm

Thus, the length of side c is approximately 8.426 cm.

Example of Law of Cosines (SSS)

Consider a triangle with sides a = 7 cm, b = 9 cm, and c = 5 cm. Find angle A.

Step 1: Identify the knowns.
- Side a = 7 cm
- Side b = 9 cm
- Side c = 5 cm
Step 2: Use the Law of Cosines.
```
a² = b² + c² - 2bc * cos(A)
```
Step 3: Rearrange the formula to solve for cos(A).
```
cos(A) = (b² + c² - a²) / (2bc)
```

Step 4: Substitute the values and solve for cos(A).

cos(A) = (9² + 5² - 7²) / (2 * 9 * 5)
cos(A) = (81 + 25 - 49) / 90
cos(A) = 57 / 90
cos(A) ≈ 0.6333

Step 5: Find angle A.
```
A = arccos(0.6333)
A ≈ 50.70°
```
Thus, angle A is approximately 50.70°.

Key Differences and Decision-Making

To effectively decide between using the Law of Sines or the Law of Cosines, consider the following table summarizing their best uses:

Scenario	Law to Use	Known Information	Goal
Angle-Side-Angle (ASA)	Law of Sines	Two angles and the included side	Find the remaining sides
Angle-Angle-Side (AAS)	Law of Sines	Two angles and a non-included side	Find the remaining sides
Side-Side-Angle (SSA)	Law of Sines	Two sides and an angle opposite one of them	Find the angle opposite the other side (Ambiguous case)
Side-Angle-Side (SAS)	Law of Cosines	Two sides and the included angle	Find the third side
Side-Side-Side (SSS)	Law of Cosines	All three sides	Find any angle

Practical Tips for Choosing the Right Law

Identify Knowns: Start by clearly identifying what information is given: sides, angles, or both.
Look for Angle-Side Pairs: If you have an angle and its opposite side, the Law of Sines is often a good choice.
Consider the Ambiguous Case: If you are given SSA, be cautious and check for multiple possible solutions.
When in Doubt, Draw a Diagram: A visual representation can help clarify the relationships between sides and angles, making it easier to choose the appropriate law.
Avoid Assumptions: Do not assume a triangle is a right triangle unless explicitly stated. The Law of Sines and Law of Cosines apply to all triangles.

Advanced Considerations

Area of a Triangle

Both the Law of Sines and the Law of Cosines can be used to find the area of a triangle.

Using the Law of Sines: If you know two sides and the included angle, you can use the formula:
```
Area = 0.5 * a * b * sin(C)
```
Using Heron's Formula (derived from the Law of Cosines): If you know all three sides, you can use Heron's formula:
```
s = (a + b + c) / 2  (where s is the semi-perimeter)
Area = √(s * (s - a) * (s - b) * (s - c))
```

Solving Real-World Problems

The Law of Sines and the Law of Cosines are not just theoretical concepts; they have numerous practical applications in fields such as:

Navigation: Determining distances and directions.
Surveying: Measuring land and creating maps.
Engineering: Designing structures and calculating forces.
Astronomy: Calculating distances between celestial bodies.

Example: Navigation Problem

A ship sails 40 miles east and then turns 30° northward and sails another 30 miles. How far is the ship from its starting point?

Step 1: Visualize the problem. The ship's path forms a triangle where two sides are known (40 miles and 30 miles) and the included angle is 150° (180° - 30°).
Step 2: Identify the knowns.
- Side a = 40 miles
- Side b = 30 miles
- Angle C = 150°

Step 3: Use the Law of Cosines to find the distance (side c).

c² = a² + b² - 2ab * cos(C)
c² = 40² + 30² - 2 * 40 * 30 * cos(150°)
c² = 1600 + 900 - 2400 * (-0.8660)
c² ≈ 2500 + 2078.4
c² ≈ 4578.4
c ≈ √4578.4
c ≈ 67.66 miles

Thus, the ship is approximately 67.66 miles from its starting point.

Common Pitfalls to Avoid

Incorrectly Identifying SSA: Always double-check for the ambiguous case when using the Law of Sines with SSA.
Misapplying Formulas: Ensure you are using the correct form of the Law of Cosines based on the information you have.
Rounding Errors: Avoid premature rounding, as it can lead to significant inaccuracies in your final answer.
Forgetting Units: Always include units in your final answer to provide context.
Assuming Right Triangles: Remember that the Law of Sines and Law of Cosines apply to all triangles, not just right triangles. For right triangles, simpler trigonometric ratios (SOH-CAH-TOA) may be more efficient.

Conclusion

Mastering the Law of Sines and the Law of Cosines is essential for anyone studying trigonometry or working in fields that require triangle solving. By understanding the specific conditions under which each law applies, you can efficiently and accurately solve a wide range of problems. Remember to carefully analyze the given information, consider potential ambiguous cases, and practice applying these laws to various scenarios. With a solid understanding and diligent practice, you'll be well-equipped to tackle any triangle-related challenge.