Unit 12 Trigonometry Homework 5 Law Of Sines Answers

Trigonometry unveils the fascinating relationships between angles and sides of triangles, a cornerstone of fields ranging from engineering to navigation. Among the many tools trigonometry offers, the Law of Sines stands out as a particularly elegant and useful method for solving triangles when you have certain pieces of information. Understanding the Law of Sines and its applications is crucial for mastering trigonometry, and homework assignments like "Unit 12 Trigonometry Homework 5: Law of Sines Answers" often test this understanding. Let's delve into this law, explore its mechanics, and work through example problems.

Understanding the Law of Sines

At its core, the Law of Sines establishes a relationship between the sides of a triangle and the sines of their opposite angles. More formally, for any triangle ABC, where 'a', 'b', and 'c' are the lengths of the sides opposite angles A, B, and C, respectively, the Law of Sines states:

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

This seemingly simple equation unlocks the ability to solve triangles in a variety of scenarios where you don't have all the angles and sides. The key is to have a pair of corresponding information: an angle and the side opposite it.

When to Use the Law of Sines:

The Law of Sines is particularly useful in two main situations:

• Angle-Angle-Side (AAS): You know two angles of the triangle and one side not between those angles.
• Angle-Side-Angle (ASA): You know two angles of the triangle and the side between those angles.
• Side-Side-Angle (SSA): You know two sides of the triangle and an angle not between those sides (this case is also known as the ambiguous case).

Let's look more closely at each of these scenarios.

AAS and ASA: Straightforward Applications

When dealing with AAS or ASA, the Law of Sines provides a direct path to finding the missing sides.

Example 1: AAS

Suppose you have a triangle where:

• Angle A = 30 degrees
• Angle B = 70 degrees
• Side a = 8 units

You want to find side 'b'.

1. Set up the Law of Sines: a / sin(A) = b / sin(B)
2. Plug in the known values: 8 / sin(30) = b / sin(70)
3. Solve for 'b': b = (8 * sin(70)) / sin(30)
4. Calculate: b ≈ (8 * 0.9397) / 0.5 ≈ 15.03 units

Therefore, side 'b' is approximately 15.03 units long. To find side 'c', you would first need to find angle C (since the angles of a triangle add up to 180 degrees). Angle C = 180 - 30 - 70 = 80 degrees. Then:

5. Set up the Law of Sines: a / sin(A) = c / sin(C)
6. Plug in the known values: 8 / sin(30) = c / sin(80)
7. Solve for 'c': c = (8 * sin(80)) / sin(30)
8. Calculate: c ≈ (8 * 0.9848) / 0.5 ≈ 15.76 units

Therefore, side 'c' is approximately 15.76 units long.

Example 2: ASA

Suppose you have a triangle where:

• Angle A = 45 degrees
• Angle C = 60 degrees
• Side b = 12 units

You want to find side 'a'.

1. Find Angle B: Angle B = 180 - 45 - 60 = 75 degrees
2. Set up the Law of Sines: a / sin(A) = b / sin(B)
3. Plug in the known values: a / sin(45) = 12 / sin(75)
4. Solve for 'a': a = (12 * sin(45)) / sin(75)
5. Calculate: a ≈ (12 * 0.7071) / 0.9659 ≈ 8.79 units

Therefore, side 'a' is approximately 8.79 units long. You can use the same process to find side 'c'.

The Ambiguous Case (SSA): A Word of Caution

The Side-Side-Angle (SSA) case, also known as the ambiguous case, requires careful attention because it can lead to zero, one, or two possible triangles. This ambiguity arises because the given side opposite the given angle might swing freely, intersecting the base in either one or two locations, or not at all.

Possible Scenarios in SSA:

• No Triangle: The given side 'a' is too short to reach the base 'c'.
• One Triangle:
  • Side 'a' is equal to the height of the triangle.
  • Side 'a' is longer than side 'c'.
• Two Triangles: Side 'a' is shorter than side 'c' but longer than the height of the triangle.

How to Handle the Ambiguous Case:

1. Calculate the Height (h): h = b * sin(A) (where 'b' is the other given side and 'A' is the given angle).
2. Compare 'a' to 'h' and 'b':
   • If a < h: No triangle exists.
   • If a = h: One right triangle exists.
   • If h < a < b: Two triangles exist.
   • If a ≥ b: One triangle exists.
3. If Two Triangles Exist: You'll need to solve for both possible angles B and, consequently, the remaining angles and sides for each triangle.

Example 3: SSA - Two Triangles

Suppose you have a triangle where:

• Angle A = 30 degrees
• Side a = 7 units
• Side b = 10 units

1. Calculate the Height: h = 10 * sin(30) = 10 * 0.5 = 5
2. Compare: Since 5 < 7 < 10 (h < a < b), there are two possible triangles.
3. Solve for Angle B (First Triangle):
   • sin(B) = (b * sin(A)) / a = (10 * sin(30)) / 7 = (10 * 0.5) / 7 ≈ 0.7143
   • B = arcsin(0.7143) ≈ 45.58 degrees
4. Solve for Angle C (First Triangle):
   • C = 180 - A - B = 180 - 30 - 45.58 ≈ 104.42 degrees
5. Solve for Side c (First Triangle):
   • c = (a * sin(C)) / sin(A) = (7 * sin(104.42)) / sin(30) ≈ (7 * 0.9685) / 0.5 ≈ 13.56 units
6. Solve for Angle B (Second Triangle): Since sine is positive in both the first and second quadrants, we need to find the supplementary angle to the first solution of angle B.
   • B' = 180 - B = 180 - 45.58 ≈ 134.42 degrees
7. Solve for Angle C (Second Triangle):
   • C' = 180 - A - B' = 180 - 30 - 134.42 ≈ 15.58 degrees
8. Solve for Side c (Second Triangle):
   • c' = (a * sin(C')) / sin(A) = (7 * sin(15.58)) / sin(30) ≈ (7 * 0.2685) / 0.5 ≈ 3.76 units

Therefore, there are two possible triangles:

• Triangle 1: A = 30°, B ≈ 45.58°, C ≈ 104.42°, a = 7, b = 10, c ≈ 13.56
• Triangle 2: A = 30°, B' ≈ 134.42°, C' ≈ 15.58°, a = 7, b = 10, c' ≈ 3.76

This example clearly demonstrates the importance of carefully analyzing the SSA case. Ignoring the possibility of two triangles can lead to an incomplete or incorrect solution.

Common Mistakes and How to Avoid Them

When working with the Law of Sines, several common mistakes can arise. Being aware of these pitfalls will help you improve your accuracy:

• Forgetting the Ambiguous Case (SSA): As highlighted above, always check for the possibility of two triangles when given SSA information.
• Using the Law of Sines Incorrectly: Make sure you have a corresponding angle-side pair to start with. You can't use the Law of Sines if you only have three sides or three angles.
• Calculator Errors: Ensure your calculator is in degree mode (if the angles are given in degrees) or radian mode (if the angles are given in radians). A simple mode error can lead to drastically wrong answers.
• Rounding Errors: Avoid rounding intermediate calculations. Round only your final answer to the specified degree of accuracy.
• Mislabeling Sides and Angles: Double-check that you've correctly assigned sides and angles according to standard notation (side 'a' is opposite angle A, and so on).
• Incorrectly Calculating the Height in SSA: Make sure you are using the correct side ('b' in our formula above) and angle ('A' in our formula above) when calculating the height to determine the number of possible triangles.

Law of Sines vs. Law of Cosines: Knowing Which to Use

The Law of Sines is a powerful tool, but it's not always the best choice for solving triangles. The Law of Cosines provides an alternative method, and knowing when to use each is crucial.

Law of Cosines: For any triangle ABC, the Law of Cosines states:

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

When to Use the Law of Cosines:

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

Key Differences and Choosing the Right Law:

In summary:

• If you have an angle-side pair, the Law of Sines is usually the faster method (but remember the ambiguous case!).
• If you don't have an angle-side pair, you'll need to use the Law of Cosines.

Real-World Applications of the Law of Sines

The Law of Sines isn't just an abstract mathematical concept; it has practical applications in various fields:

• Surveying: Surveyors use the Law of Sines to calculate distances and angles in land measurement.
• Navigation: Sailors and pilots rely on the Law of Sines to determine their position and heading.
• Engineering: Engineers use the Law of Sines in structural analysis and design.
• Astronomy: Astronomers use trigonometric principles, including the Law of Sines, to calculate distances to stars and planets.
• Forensic Science: Crime scene investigators can use the Law of Sines to reconstruct events based on angles and distances.

Example Problems and Solutions

Here are some additional example problems to solidify your understanding of the Law of Sines:

Problem 1:

In triangle PQR, angle P = 110°, angle Q = 35°, and side r = 15 cm. Find side p.

Solution:

1. Find Angle R: R = 180 - 110 - 35 = 35°
2. Set up the Law of Sines: p / sin(P) = r / sin(R)
3. Plug in the values: p / sin(110) = 15 / sin(35)
4. Solve for p: p = (15 * sin(110)) / sin(35)
5. Calculate: p ≈ (15 * 0.9397) / 0.5736 ≈ 24.58 cm

Problem 2:

In triangle XYZ, side x = 10 m, side y = 8 m, and angle X = 60°. Find angle Y.

Solution:

1. Set up the Law of Sines: sin(Y) / y = sin(X) / x
2. Plug in the values: sin(Y) / 8 = sin(60) / 10
3. Solve for sin(Y): sin(Y) = (8 * sin(60)) / 10 = (8 * 0.8660) / 10 ≈ 0.6928
4. Solve for Y: Y = arcsin(0.6928) ≈ 43.80°
5. Check for a Second Triangle: Since x > y, there is only one possible triangle.

Problem 3:

In triangle ABC, angle A = 40°, side a = 6 cm, and side b = 8 cm. Determine how many triangles are possible.

Solution:

1. Calculate the Height: h = b * sin(A) = 8 * sin(40) ≈ 8 * 0.6428 ≈ 5.14 cm
2. Compare: Since h < a < b (5.14 < 6 < 8), there are two possible triangles. You would then proceed to solve for both triangles as demonstrated in Example 3.

Tips for Success on "Unit 12 Trigonometry Homework 5"

When tackling trigonometry homework assignments focusing on the Law of Sines, consider these tips:

• Draw Diagrams: Always sketch a diagram of the triangle. This helps visualize the problem and correctly identify sides and angles.
• Label Carefully: Label all known sides and angles on your diagram.
• Choose the Right Law: Determine whether the Law of Sines or the Law of Cosines is appropriate based on the given information.
• Show Your Work: Clearly show each step of your calculations. This allows you to identify and correct errors more easily.
• Check Your Answers: Make sure your answers are reasonable within the context of the problem. For example, the largest angle should be opposite the longest side.
• Pay Attention to Units: Include the correct units in your final answers (e.g., cm, m, degrees).
• Review Examples: Rework example problems from your textbook or class notes.
• Seek Help When Needed: Don't hesitate to ask your teacher or classmates for help if you're struggling with a particular problem.

By mastering the Law of Sines and understanding its nuances, you'll be well-equipped to tackle a wide range of trigonometry problems and appreciate its practical applications in the real world. Remember to practice consistently, pay attention to detail, and always double-check your work. Good luck with your "Unit 12 Trigonometry Homework 5"!