When To Use Sine Law And Cosine Law

Diving into the world of trigonometry often brings about a crucial question for students: when exactly do you use the sine law and when should you apply the cosine law? Mastering this distinction is essential for tackling a wide array of problems involving triangles, from calculating distances in surveying to understanding forces in physics. This comprehensive guide will walk you through the intricacies of both laws, providing clear guidelines and examples to help you confidently choose the right tool for the job.

Understanding the Sine Law

The sine law, also known as the law of sines, establishes a relationship between the angles of a triangle and the lengths of their opposite sides. It's a powerful tool, particularly useful when you have certain combinations of angles and sides known.

The Formula

The sine law is expressed as follows:

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 Sine Law

The sine law is your go-to method when you have one of the following scenarios:

1. Angle-Side-Angle (ASA): You know two angles and the included side (the side between the two angles).
2. Angle-Angle-Side (AAS): You know two angles and a non-included side.
3. Side-Side-Angle (SSA): You know two sides and an angle opposite one of those sides. This case is also known as the ambiguous case because it may result in zero, one, or two possible solutions.

Let's delve deeper into each scenario with illustrative examples.

Scenario 1: Angle-Side-Angle (ASA)

Imagine a surveyor needs to determine the distance across a river. They measure an angle of 60 degrees from point A to a tree on the opposite bank (point C). They then walk 100 meters along the riverbank to point B and measure the angle to the same tree as 45 degrees. How far is point A from the tree (point C)?

• Known: Angle A = 60°, Angle B = 45°, side AB (c) = 100 meters.
• Goal: Find side b (distance from A to C).

First, calculate angle C: C = 180° - A - B = 180° - 60° - 45° = 75°.

Now, apply the sine law:

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

b / sin(45°) = 100 / sin(75°)

b = (100 * sin(45°)) / sin(75°)

b ≈ 73.2 meters

Therefore, the distance from point A to the tree (point C) is approximately 73.2 meters.

Scenario 2: Angle-Angle-Side (AAS)

Consider a situation where an airplane is spotted from two ground stations that are 2 miles apart. Station A reports the angle of elevation to the plane is 35°, while Station B reports an angle of elevation of 50°. What is the altitude of the airplane if it's directly above the line connecting the two stations?

• Known: Angle A = 35°, Angle B = 50°, side AB (c) = 2 miles.
• Goal: Find the altitude (h) of the airplane.

First, we need to find angle C: C = 180° - 35° - 50° = 95°.

Next, let’s find side a (distance from Station B to the airplane):

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

a / sin(35°) = 2 / sin(95°)

a = (2 * sin(35°)) / sin(95°)

a ≈ 1.15 miles

Now, we can use basic trigonometry (sine function) to find the altitude (h):

sin(B) = h / a

sin(50°) = h / 1.15

h = 1.15 * sin(50°)

h ≈ 0.88 miles

The altitude of the airplane is approximately 0.88 miles.

Scenario 3: Side-Side-Angle (SSA) - The Ambiguous Case

This scenario requires careful consideration because it can lead to multiple possible triangles, one triangle, or no triangle at all. Let's illustrate with an example.

Suppose we have a triangle where side a = 15 cm, side b = 20 cm, and angle A = 30°. Find angle B.

• Known: a = 15 cm, b = 20 cm, A = 30°
• Goal: Find angle B

Using the sine law:

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

sin(B) / 20 = sin(30°) / 15

sin(B) = (20 * sin(30°)) / 15

sin(B) ≈ 0.6667

Now, we find the principal value of B:

B₁ = arcsin(0.6667) ≈ 41.8°

However, because the sine function is positive in both the first and second quadrants, there might be another possible angle:

B₂ = 180° - 41.8° ≈ 138.2°

Now we need to check if both angles are valid. We do this by checking if the sum of A and each possible B is less than 180°.

• For B₁: A + B₁ = 30° + 41.8° = 71.8° < 180° (Valid)
• For B₂: A + B₂ = 30° + 138.2° = 168.2° < 180° (Valid)

Since both angles are valid, there are two possible triangles that satisfy the given conditions. This is why SSA is called the ambiguous case.

Understanding the Cosine Law

The cosine law, also known as the law of cosines, relates the lengths of the sides of a triangle to the cosine of one of its angles. It is essential when you don't have a matching angle-side pair, making it a perfect alternative to the sine law in certain situations.

The Formula

The cosine law can be expressed in three different forms, each focusing on 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 Cosine Law

The cosine law is your go-to method when you have one of the following scenarios:

1. Side-Angle-Side (SAS): You know two sides and the included angle (the angle between the two sides).
2. Side-Side-Side (SSS): You know all three sides of the triangle.

Let’s explore each scenario with examples.

Scenario 1: Side-Angle-Side (SAS)

Imagine a triangle where side a = 5 cm, side b = 8 cm, and angle C (the angle between sides a and b) is 77°. Find the length of side c.

• Known: a = 5 cm, b = 8 cm, C = 77°
• Goal: Find side c

Using the cosine law:

c² = a² + b² - 2ab * cos(C)

c² = 5² + 8² - 2 * 5 * 8 * cos(77°)

c² = 25 + 64 - 80 * cos(77°)

c² ≈ 25 + 64 - 80 * 0.225

c² ≈ 90 - 18

c² ≈ 72

c ≈ √72

c ≈ 8.49 cm

Therefore, the length of side c is approximately 8.49 cm.

Scenario 2: Side-Side-Side (SSS)

Consider a triangle with sides a = 7 m, b = 9 m, and c = 12 m. Find the measure of angle A.

• Known: a = 7 m, b = 9 m, c = 12 m
• Goal: Find angle A

Using the cosine law, rearranged to solve for the angle:

a² = b² + c² - 2bc * cos(A)

cos(A) = (b² + c² - a²) / (2bc)

cos(A) = (9² + 12² - 7²) / (2 * 9 * 12)

cos(A) = (81 + 144 - 49) / 216

cos(A) = 176 / 216

cos(A) ≈ 0.8148

A = arccos(0.8148)

A ≈ 35.4°

Therefore, the measure of angle A is approximately 35.4 degrees.

Key Differences and How to Choose

The fundamental difference between the sine law and the cosine law lies in the information you have about the triangle.

• Sine Law: Use when you have a matching pair of an angle and its opposite side, plus one additional piece of information (either another angle or another side).
• Cosine Law: Use when you don't have a matching pair. Specifically, use it when you have two sides and the included angle (SAS) or all three sides (SSS).

Here’s a quick reference table:

Law	When to Use	Information Required
Sine Law	When you have an angle and its opposite side.	ASA, AAS, SSA (be cautious of the ambiguous case with SSA).
Cosine Law	When you don't have a matching angle-side pair.	SAS, SSS

Practical Applications

Both the sine and cosine laws have wide-ranging applications in various fields:

• Surveying: Determining distances and angles in land measurement.
• Navigation: Calculating distances and bearings in air and sea navigation.
• Engineering: Analyzing forces and stresses in structures.
• Physics: Solving problems involving vectors and mechanics.
• Astronomy: Measuring distances to stars and planets.

Common Mistakes to Avoid

• Confusing ASA and AAS: Be precise about which sides and angles are given. ASA means the side is between the two angles, while AAS means it is not.
• Forgetting the Ambiguous Case (SSA): Always check for a second possible solution when using the sine law with SSA. Calculate the supplementary angle and see if it forms a valid triangle.
• Incorrectly Rearranging the Cosine Law: Make sure you are using the correct form of the cosine law to isolate the angle you are trying to find.
• Using the Wrong Law: Double-check the information you have before deciding which law to apply. If you have a matching angle-side pair, consider the sine law first. If not, the cosine law is likely the correct choice.
• Calculator Settings: Ensure your calculator is set to the correct mode (degrees or radians) depending on the units used in the problem.

Examples Combining Sine and Cosine Laws

Sometimes, a problem might require you to use both the sine and cosine laws in sequence. Here’s an example:

Consider a triangle ABC where AB = 10 cm, BC = 12 cm, and angle B = 70°.

1. Find AC (side b): Since we have SAS (Side-Angle-Side), we use the cosine law:

   b² = a² + c² - 2ac * cos(B)

   b² = 12² + 10² - 2 * 12 * 10 * cos(70°)

   b² = 144 + 100 - 240 * cos(70°)

   b² ≈ 144 + 100 - 240 * 0.342

   b² ≈ 244 - 82.08

   b² ≈ 161.92

   b ≈ √161.92

   b ≈ 12.72 cm

2. Find angle A: Now that we know all three sides, we could use the cosine law again to find angle A. However, we also now have a matching angle-side pair (angle B and side b), so we can use the sine law:

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

   sin(A) / 12 = sin(70°) / 12.72

   sin(A) = (12 * sin(70°)) / 12.72

   sin(A) ≈ (12 * 0.94) / 12.72

   sin(A) ≈ 0.888

   A ≈ arcsin(0.888)

   A ≈ 62.6°

3. Find angle C: Angle C can be found using the fact that the sum of angles in a triangle is 180°:

   C = 180° - A - B

   C = 180° - 62.6° - 70°

   C ≈ 47.4°

In this example, we used the cosine law to find the missing side and then switched to the sine law to find a missing angle.

Practice Problems

To solidify your understanding, try solving these problems. Remember to identify whether you have ASA, AAS, SSA, SAS, or SSS, and then choose the appropriate law.

1. In triangle PQR, angle P = 110°, angle Q = 30°, and side p = 15 cm. Find side q.
2. In triangle XYZ, side x = 8 m, side y = 5 m, and angle Z = 60°. Find side z.
3. In triangle ABC, side a = 6 ft, side b = 8 ft, and side c = 12 ft. Find angle A.
4. In triangle DEF, angle D = 25°, side e = 10 inches, and side d = 6 inches. Find angle E (be mindful of the ambiguous case).

Conclusion

Mastering the sine and cosine laws is a fundamental step in trigonometry. By understanding the conditions under which each law applies, you can confidently solve a wide range of problems involving triangles. Remember to carefully analyze the given information, choose the appropriate law, and be mindful of potential pitfalls like the ambiguous case. With practice and a solid understanding of these concepts, you’ll be well-equipped to tackle even the most challenging trigonometric problems.