Homework 7 Law Of Sines Answers

The Law of Sines is a powerful tool in trigonometry that allows us to solve triangles when we don't have a right angle. It establishes a relationship between the sides of a triangle and the sines of their opposite angles. Understanding and applying this law is crucial for various applications in fields like surveying, navigation, and engineering. Mastering homework assignments focusing on the Law of Sines requires a solid grasp of the formula, its applications, and common problem-solving strategies.

Understanding the Law of Sines

The Law of Sines states that for any triangle ABC, with sides a, b, and c opposite to angles A, B, and C respectively, the following relationship holds:

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

This formula allows us to find unknown sides or angles of a triangle if we have enough information. Typically, we need to know:

Two angles and one side (AAS or ASA): This allows us to find the remaining angle (since the angles in a triangle sum to 180 degrees) and the other two sides.
Two sides and an angle opposite one of them (SSA): This is known as the ambiguous case, as it can lead to zero, one, or two possible solutions. We'll explore this case in more detail later.

Key Concepts and Formulas

Angle-Side Relationship: The Law of Sines highlights the direct relationship between the length of a side and the sine of its opposite angle. Larger angles correspond to longer sides, and vice versa.
Sine Function: A fundamental trigonometric function that relates an angle to the ratio of the opposite side to the hypotenuse in a right-angled triangle. While the Law of Sines applies to non-right triangles, understanding the sine function is essential.
Triangle Angle Sum Theorem: The sum of the interior angles of any triangle is always 180 degrees. This theorem is crucial for finding missing angles.

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

Here's a breakdown of how to approach and solve problems involving the Law of Sines:

1. Identify the Given Information:

Carefully read the problem statement and identify which angles and sides are given.
Draw a diagram of the triangle and label the known quantities. This visual representation can help you understand the problem better.
Determine if you have AAS, ASA, or SSA.

2. Apply the Law of Sines:

Set up the Law of Sines equation using the known values. Choose the two ratios that involve the known quantities and the unknown quantity you want to find.
For example, if you know angle A, angle B, and side a, and you want to find side b, you would set up the equation:
```
a / sin(A) = b / sin(B)
```

3. Solve for the Unknown:

Use algebraic manipulation to isolate the unknown variable.
Cross-multiply to eliminate the fractions:
```
a * sin(B) = b * sin(A)
```
Divide both sides by sin(A) to solve for b:
```
b = (a * sin(B)) / sin(A)
```

4. Calculate the Result:

Use a calculator to evaluate the trigonometric functions (sine) and perform the arithmetic calculations.
Ensure your calculator is in the correct mode (degrees or radians) depending on the units of the given angles.

5. Check Your Answer:

Does the answer make sense in the context of the problem? For example, is the calculated side length positive and reasonable compared to the other side lengths?
If you found an angle, is it within the valid range of 0 to 180 degrees?
Verify that the largest angle is opposite the longest side, and the smallest angle is opposite the shortest side.

Example Problems and Solutions

Let's work through some example problems to illustrate the application of the Law of Sines:

Example 1 (AAS):

In triangle ABC, angle A = 30 degrees, angle B = 70 degrees, and side a = 8 cm. Find side b.

Given: A = 30°, B = 70°, a = 8 cm
Find: b

Solution:

Apply the Law of Sines:
```
a / sin(A) = b / sin(B)
```
Substitute the given values:
```
8 / sin(30°) = b / sin(70°)
```

Solve for b:

b = (8 * sin(70°)) / sin(30°)
b = (8 * 0.9397) / 0.5
b = 15.035 cm

Answer: Side b is approximately 15.04 cm.

Example 2 (ASA):

In triangle PQR, angle P = 40 degrees, angle Q = 60 degrees, and side r = 12 meters. Find side p.

Given: P = 40°, Q = 60°, r = 12 m
Find: p

Solution:

First, find angle R using the Triangle Angle Sum Theorem:
```
R = 180° - P - Q
R = 180° - 40° - 60°
R = 80°
```
Now, apply the Law of Sines:
```
p / sin(P) = r / sin(R)
```
Substitute the given values:
```
p / sin(40°) = 12 / sin(80°)
```

Solve for p:

p = (12 * sin(40°)) / sin(80°)
p = (12 * 0.6428) / 0.9848
p = 7.83 m

Answer: Side p is approximately 7.83 meters.

Example 3 (Finding an Angle):

In triangle XYZ, side x = 10 inches, side y = 15 inches, and angle X = 25 degrees. Find angle Y.

Given: x = 10 in, y = 15 in, X = 25°
Find: Y
Solution:
- Apply the Law of Sines:
```
x / sin(X) = y / sin(Y)
```
- Substitute the given values:
```
10 / sin(25°) = 15 / sin(Y)
```
- Solve for sin(Y):
```
sin(Y) = (15 * sin(25°)) / 10
sin(Y) = (15 * 0.4226) / 10
sin(Y) = 0.6339
```
- Find angle Y using the inverse sine function (arcsin):
```
Y = arcsin(0.6339)
Y = 39.34°
```
- Answer: Angle Y is approximately 39.34 degrees.

The Ambiguous Case (SSA)

The SSA (Side-Side-Angle) case is called "ambiguous" because the given information might lead to zero, one, or two possible triangles. This ambiguity arises because the given angle is not included between the two given sides. Here's how to analyze the SSA case:

1. Calculate h (the height):

h = b * sin(A), where A is the given angle, and b is the side adjacent to angle A.

2. Compare a (the side opposite angle A) with h and b:

Case 1: a < h (No Triangle)
- There is no triangle possible because side a is too short to reach the base.
Case 2: a = h (One Triangle - Right Triangle)
- Side a is exactly the height, forming a right triangle.
Case 3: h < a < b (Two Triangles)
- Side a can swing in two different positions to form two distinct triangles. You will need to find two possible solutions for the unknown angle.
Case 4: a ≥ b (One Triangle)
- Side a is long enough to reach the base in only one possible configuration.

Example of the Ambiguous Case:

In triangle ABC, angle A = 30 degrees, side b = 12 cm, and side a = 7 cm. How many triangles are possible? Find angle B.

Given: A = 30°, b = 12 cm, a = 7 cm
Find: Number of possible triangles, angle B
Solution:
1. Calculate h:
```
h = b * sin(A)
h = 12 * sin(30°)
h = 12 * 0.5
h = 6 cm
```
2. Compare a with h and b:
  - We have h = 6, a = 7, and b = 12.
  - Since h < a < b (6 < 7 < 12), there are two possible triangles.
3. Find angle B:
  - Apply the Law of Sines:
```
a / sin(A) = b / sin(B)
```
  - Substitute the given values:
```
7 / sin(30°) = 12 / sin(B)
```
  - Solve for sin(B):
```
sin(B) = (12 * sin(30°)) / 7
sin(B) = (12 * 0.5) / 7
sin(B) = 0.8571
```
  - Find angle B using the inverse sine function (arcsin):
```
B₁ = arcsin(0.8571)
B₁ = 59.0°
```
  - Since there are two possible triangles, we need to find the second possible angle B (B₂). Because sin(x) = sin(180° - x), we can find the second angle by subtracting B₁ from 180 degrees:
```
B₂ = 180° - B₁
B₂ = 180° - 59.0°
B₂ = 121.0°
```
  - We need to check if both B₁ and B₂ are valid. A valid angle B will result in a valid angle C (180° - A - B > 0).
    - For B₁ = 59.0°:
```
C₁ = 180° - A - B₁
C₁ = 180° - 30° - 59.0°
C₁ = 91.0° (Valid)
```
    - For B₂ = 121.0°:
```
C₂ = 180° - A - B₂
C₂ = 180° - 30° - 121.0°
C₂ = 29.0° (Valid)
```
- Answer: There are two possible triangles. The two possible values for angle B are approximately 59.0 degrees and 121.0 degrees.

Common Mistakes and How to Avoid Them

Calculator Mode: Ensure your calculator is in the correct mode (degrees or radians). A small mistake here can lead to drastically incorrect answers. Double-check the problem to see if the angles are given in degrees or radians.
Ambiguous Case Neglect: Forgetting to check for the ambiguous case (SSA) can lead to missing a solution or providing an incorrect one. Always calculate h and compare it to the given sides.
Incorrect Setup: Misidentifying the angle-side relationships can lead to an incorrect setup of the Law of Sines equation. Draw a clear diagram and label the sides and angles correctly.
Rounding Errors: Rounding intermediate calculations too early can accumulate errors and affect the final answer. Keep as many decimal places as possible during calculations and round only at the very end.
Missing the Second Triangle in SSA: In the ambiguous case, if two triangles are possible, remember to find both solutions for the unknown angle using the supplementary angle property (sin(x) = sin(180° - x)).
Assuming Right Triangle Applicability: The Law of Sines can be used on right triangles, but it's generally more straightforward to use basic trigonometric ratios (SOH CAH TOA) in those cases. The Law of Sines is particularly useful for non-right triangles.

Advanced Applications and Extensions

The Law of Sines is not just a theoretical concept; it has practical applications in various fields:

Surveying: Surveyors use the Law of Sines to calculate distances and angles in land measurement and mapping.
Navigation: Sailors and pilots use the Law of Sines to determine their position and course.
Engineering: Engineers use the Law of Sines in structural analysis and design, particularly when dealing with triangular structures.
Astronomy: Astronomers use the Law of Sines to calculate distances to stars and other celestial objects.

Extensions:

Law of Cosines: When the Law of Sines is not sufficient (e.g., in the SSS or SAS cases), the Law of Cosines can be used to solve triangles.
Area of a Triangle: The area of a triangle can be calculated using the formula: Area = (1/2) * b * c * sin(A), which involves the sine of an angle.
Vector Addition: The Law of Sines can be used in conjunction with vector addition to find the resultant vector when adding two or more vectors.

Conclusion

Mastering the Law of Sines is a crucial step in your journey through trigonometry. By understanding the fundamental concepts, practicing problem-solving techniques, and being aware of common pitfalls like the ambiguous case, you can confidently tackle a wide range of trigonometry problems. Remember to draw diagrams, carefully label known and unknown quantities, and check your answers for reasonableness. This will pave the way for success in your homework assignments and beyond, opening doors to the exciting applications of trigonometry in various fields.