Law Of Sine Law Of Cosine

The Law of Sines and the Law of Cosines are fundamental trigonometric principles that extend the use of sine and cosine beyond right triangles. They allow us to solve for unknown sides and angles in any triangle, regardless of whether it contains a right angle. This article delves into these laws, explaining their applications, providing examples, and elucidating their derivations.

Introduction to Oblique Triangles

Before diving into the laws, it’s crucial to understand oblique triangles. These are triangles that do not contain a right angle (90 degrees). An oblique triangle can either be:

Acute: All three angles are less than 90 degrees.
Obtuse: One angle is greater than 90 degrees.

Solving oblique triangles requires tools beyond the basic trigonometric ratios (SOH CAH TOA) applicable to right triangles. This is where the Law of Sines and Law of Cosines become essential.

The Law of Sines: Unveiling Relationships

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.

Mathematical Formulation:

For a triangle ABC, where 'a' is the length of the side opposite angle A, 'b' is the length of the side opposite angle B, and 'c' is the length of the side opposite angle C, the Law of Sines is expressed as:

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

When to Use the Law of Sines:

The Law of Sines is particularly useful in the following scenarios:

Angle-Side-Angle (ASA): Given two angles and the included side.
Angle-Angle-Side (AAS): Given two angles and a non-included side.
Side-Side-Angle (SSA): Given two sides and an angle opposite one of those sides (this case can be ambiguous, leading to zero, one, or two possible solutions).

Solving Triangles Using the Law of Sines: Step-by-Step

Let's consider each case and illustrate how to apply the Law of Sines:

ASA (Angle-Side-Angle): Suppose we have a triangle where angle A = 30°, angle C = 70°, and side b = 10. We want to find the remaining angles and sides.
1. Find angle B: Since the angles in a triangle sum to 180°, B = 180° - A - C = 180° - 30° - 70° = 80°.
2. Apply the Law of Sines:
  - a / sin(A) = b / sin(B) => a / sin(30°) = 10 / sin(80°) => a = (10 * sin(30°)) / sin(80°) ≈ 5.08
  - c / sin(C) = b / sin(B) => c / sin(70°) = 10 / sin(80°) => c = (10 * sin(70°)) / sin(80°) ≈ 9.54
Therefore, we have found all angles and sides: A = 30°, B = 80°, C = 70°, a ≈ 5.08, b = 10, c ≈ 9.54.
AAS (Angle-Angle-Side): Suppose we have a triangle where angle A = 40°, angle B = 60°, and side a = 8. We want to find the remaining angles and sides.
1. Find angle C: Since the angles in a triangle sum to 180°, C = 180° - A - B = 180° - 40° - 60° = 80°.
2. Apply the Law of Sines:
  - b / sin(B) = a / sin(A) => b / sin(60°) = 8 / sin(40°) => b = (8 * sin(60°)) / sin(40°) ≈ 10.77
  - c / sin(C) = a / sin(A) => c / sin(80°) = 8 / sin(40°) => c = (8 * sin(80°)) / sin(40°) ≈ 12.23
Therefore, we have found all angles and sides: A = 40°, B = 60°, C = 80°, a = 8, b ≈ 10.77, c ≈ 12.23.
SSA (Side-Side-Angle): The Ambiguous Case: This is the trickiest scenario. Given two sides and an angle opposite one of them, there might be zero, one, or two possible triangles. Let's illustrate with an example and then discuss the different possibilities.

Suppose we have a triangle where a = 10, b = 12, and A = 30°. We want to find the remaining angles and sides.
1. Apply the Law of Sines:
  - sin(B) / b = sin(A) / a => sin(B) / 12 = sin(30°) / 10 => sin(B) = (12 * sin(30°)) / 10 = 0.6
2. Find angle B:
  - B = arcsin(0.6) ≈ 36.87°
3. Check for a second possible solution: Since sin(x) = sin(180° - x), there might be another possible angle for B:
  - B' = 180° - 36.87° ≈ 143.13°
4. Determine if both solutions are valid: We need to check if both angles B and B', when added to angle A, result in a sum less than 180°.
  - A + B = 30° + 36.87° = 66.87° < 180° (Valid)
  - A + B' = 30° + 143.13° = 173.13° < 180° (Valid)
Since both angles are valid, we have two possible triangles. Let's solve for each:

Triangle 1 (using B ≈ 36.87°):
- C = 180° - A - B = 180° - 30° - 36.87° ≈ 113.13°
- c / sin(C) = a / sin(A) => c / sin(113.13°) = 10 / sin(30°) => c = (10 * sin(113.13°)) / sin(30°) ≈ 18.47
Triangle 2 (using B' ≈ 143.13°):
- C' = 180° - A - B' = 180° - 30° - 143.13° ≈ 6.87°
- c' / sin(C') = a / sin(A) => c' / sin(6.87°) = 10 / sin(30°) => c' = (10 * sin(6.87°)) / sin(30°) ≈ 2.39
Therefore, in the SSA case, we found two possible solutions:
- Triangle 1: A = 30°, B ≈ 36.87°, C ≈ 113.13°, a = 10, b = 12, c ≈ 18.47
- Triangle 2: A = 30°, B' ≈ 143.13°, C' ≈ 6.87°, a = 10, b = 12, c' ≈ 2.39
Understanding the Ambiguity: The ambiguity in the SSA case arises because the given side opposite the angle can potentially "swing" and intersect the base line at two different points, creating two distinct triangles that satisfy the given conditions.

Conditions for the SSA Case:
- No Solution: If sin(B) > 1, then there is no possible triangle because the sine function's range is [-1, 1]. Also, if the angle opposite the given side is obtuse and the given side is shorter than the other given side, there's no solution.
- One Solution: If the angle opposite the given side is acute and the given side is longer than the other given side, there's only one solution. Also, if sin(B) = 1, there is one right triangle solution.
- Two Solutions: If the angle opposite the given side is acute, the given side is shorter than the other given side, and the given side is long enough to reach the base line, there are two solutions (as demonstrated in the example above). The condition for this is: a < b and a > b * sin(A).

The Law of Cosines: Dealing with Side-Side-Side and Side-Angle-Side

The Law of Cosines provides a relationship between the sides and angles of a triangle, especially useful when the Law of Sines cannot be directly applied. It's particularly helpful in cases where you know all three sides or two sides and the included angle.

Mathematical Formulation:

For a triangle ABC, the Law of Cosines is expressed in three forms:

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

Notice that if angle C is 90 degrees, cos(C) = 0, and the equation c² = a² + b² - 2ab * cos(C) simplifies to the Pythagorean theorem: c² = a² + b². Thus, the Law of Cosines is a generalization of the Pythagorean theorem.

When to Use the Law of Cosines:

The Law of Cosines is most useful in the following scenarios:

Side-Angle-Side (SAS): Given two sides and the included angle.
Side-Side-Side (SSS): Given all three sides.

Solving Triangles Using the Law of Cosines: Step-by-Step

SAS (Side-Angle-Side): Suppose we have a triangle where a = 5, b = 8, and angle C = 77°. We want to find the remaining angles and sides.
1. Find side c using the Law of Cosines:
  - c² = a² + b² - 2ab * cos(C) = 5² + 8² - 2 * 5 * 8 * cos(77°) ≈ 25 + 64 - 80 * 0.225 = 71
  - c = √71 ≈ 8.43
2. Find angle A using the Law of Sines or Law of Cosines: It's generally safer to use the Law of Cosines to find angles, especially after having already used it to find a side, as the Law of Sines can sometimes lead to ambiguous results. Let's use the Law of Cosines to find angle A.
  - a² = b² + c² - 2bc * cos(A) => cos(A) = (b² + c² - a²) / (2bc) = (8² + 71 - 5²) / (2 * 8 * √71) = (64 + 71 - 25) / (16 * √71) ≈ 0.674
  - A = arccos(0.674) ≈ 47.65°
3. Find angle B:
  - B = 180° - A - C = 180° - 47.65° - 77° ≈ 55.35°
Therefore, we have found all angles and sides: A ≈ 47.65°, B ≈ 55.35°, C = 77°, a = 5, b = 8, c ≈ 8.43.
SSS (Side-Side-Side): Suppose we have a triangle where a = 7, b = 11, and c = 8. We want to find all three angles.
1. Find angle A using the Law of Cosines:
  - a² = b² + c² - 2bc * cos(A) => cos(A) = (b² + c² - a²) / (2bc) = (11² + 8² - 7²) / (2 * 11 * 8) = (121 + 64 - 49) / 176 ≈ 0.7727
  - A = arccos(0.7727) ≈ 39.48°
2. Find angle B using the Law of Cosines:
  - b² = a² + c² - 2ac * cos(B) => cos(B) = (a² + c² - b²) / (2ac) = (7² + 8² - 11²) / (2 * 7 * 8) = (49 + 64 - 121) / 112 ≈ -0.0714
  - B = arccos(-0.0714) ≈ 94.10°
3. Find angle C:
  - C = 180° - A - B = 180° - 39.48° - 94.10° ≈ 46.42°
Therefore, we have found all angles: A ≈ 39.48°, B ≈ 94.10°, C ≈ 46.42°, a = 7, b = 11, c = 8.

Derivation of the Law of Sines

The Law of Sines can be derived using the area of a triangle. Consider a triangle ABC. The area of the triangle can be expressed in three different ways using the sine function:

Area = (1/2) * b * c * sin(A)
Area = (1/2) * a * c * sin(B)
Area = (1/2) * a * b * sin(C)

Since all three expressions represent the same area, we can equate them:

(1/2) * b * c * sin(A) = (1/2) * a * c * sin(B) = (1/2) * a * b * sin(C)

Divide each term by (1/2)abc:

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

Taking the reciprocal of each term gives the Law of Sines:

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

Derivation of the Law of Cosines

The Law of Cosines can be derived using the distance formula and the properties of right triangles within a coordinate plane. Consider a triangle ABC placed in the coordinate plane with vertex C at the origin (0,0) and vertex B on the x-axis at (a, 0). Then, let vertex A have coordinates (x, y).

Using trigonometry, we can express the coordinates of A in terms of side b and angle C:

x = b * cos(C)
y = b * sin(C)

Now, we can use the distance formula to find the length of side c (the distance between A and B):

c = √[(x - a)² + (y - 0)²]

Squaring both sides:

c² = (x - a)² + y²

Substitute the expressions for x and y:

c² = (b * cos(C) - a)² + (b * sin(C))²

Expand the equation:

c² = b² * cos²(C) - 2ab * cos(C) + a² + b² * sin²(C)

Rearrange the terms:

c² = a² + b² * (cos²(C) + sin²(C)) - 2ab * cos(C)

Since cos²(C) + sin²(C) = 1:

c² = a² + b² - 2ab * cos(C)

This is the Law of Cosines. The other two forms of the law can be derived similarly by placing different vertices at the origin.

Practical Applications

The Law of Sines and Law of Cosines have numerous applications in various fields:

Surveying: Calculating distances and angles in land surveying.
Navigation: Determining the position and course of ships and aircraft.
Engineering: Designing structures and calculating forces.
Physics: Analyzing projectile motion and wave phenomena.
Astronomy: Calculating distances between stars and planets.

Example Application: Navigation

A ship sails 40 miles due east and then changes course to N 40° E and sails another 30 miles. How far is the ship from its starting point?

Visualize the problem: Draw a diagram representing the ship's path. The first leg is 40 miles east. The second leg is 30 miles at an angle of 40° north of east. The angle between the two legs is 180° - 40° = 140°.
Apply the Law of Cosines: Let 'c' be the distance from the starting point. We have a = 40, b = 30, and C = 140°.
- c² = a² + b² - 2ab * cos(C) = 40² + 30² - 2 * 40 * 30 * cos(140°)
- c² ≈ 1600 + 900 - 2400 * (-0.766) ≈ 2500 + 1838.4 ≈ 4338.4
- c ≈ √4338.4 ≈ 65.87 miles

Therefore, the ship is approximately 65.87 miles from its starting point.

Common Mistakes and How to Avoid Them

Ambiguous Case (SSA): Always check for the possibility of two solutions when using the Law of Sines in the SSA case. Calculate the supplementary angle and see if it results in a valid triangle.
Incorrect Angle Selection: Ensure you are using the correct angle opposite the corresponding side when applying both laws.
Calculator Settings: Make sure your calculator is in the correct mode (degrees or radians) depending on the problem.
Rounding Errors: Avoid rounding intermediate calculations to maintain accuracy in the final answer. Round only at the very end.
Confusing Sines and Cosines: Understand when to apply each law based on the given information. If you have ASA, AAS, or SSA (and check for ambiguity), use the Law of Sines. If you have SAS or SSS, use the Law of Cosines.

Conclusion

The Law of Sines and the Law of Cosines are powerful tools for solving oblique triangles. Mastering these laws provides a solid foundation for tackling more complex trigonometric problems in various scientific and engineering disciplines. Understanding the conditions under which each law is applicable, along with careful attention to potential ambiguities, will ensure accurate and reliable solutions. From surveying land to navigating the seas, these principles remain essential for understanding the relationships between angles and sides in any triangle.