Solve Triangles Using The Law Of Sines Khan Academy Answers

Solving triangles using the Law of Sines is a fundamental concept in trigonometry, offering a powerful method to determine unknown angles and side lengths when certain information about the triangle is provided. The Law of Sines establishes a relationship between the angles of a triangle and the lengths of their opposite sides, making it an indispensable tool for various applications in mathematics, physics, and engineering.

Understanding the Law of Sines

The Law of Sines states that for any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. Mathematically, this can be 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 to sides a, b, and c, respectively.

This formula allows us to solve for unknown sides or angles when we know at least one side-angle pair and one additional piece of information (either a side or an angle).

Conditions for Using the Law of Sines

Before diving into the mechanics of solving triangles with the Law of Sines, it's crucial to understand the scenarios where this law is applicable. The Law of Sines is primarily used in two main cases:

1. Angle-Side-Angle (ASA): When we know two angles and the included side (the side between the two angles).
2. Angle-Angle-Side (AAS): When we know two angles and a non-included side (a side that is not between the two angles).
3. Side-Side-Angle (SSA): When we know two sides and an angle opposite one of them. This case is also known as the ambiguous case because it may lead to one, two, or no possible triangles.

Steps to Solve Triangles Using the Law of Sines

Solving triangles using the Law of Sines involves a systematic approach. Here are the steps you can follow:

Step 1: Identify the Given Information

The first step is to carefully identify what information is provided in the problem. Determine which angles and sides are known. This step is critical because it dictates whether the Law of Sines can be applied and how to proceed.

Step 2: Set Up the Law of Sines Equation

Based on the given information, set up the Law of Sines equation. Choose the ratios that involve the known values and the unknown value you want to find. To give you an idea, if you know angle A, side a, and angle B, and you want to find side b, you would set up the equation as:

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

Step 3: Solve for the Unknown

Use algebraic manipulation to solve for the unknown side or angle. This usually involves cross-multiplication and the use of trigonometric functions (sine, arcsine).

Step 4: Find Missing Angles or Sides

Once you have found one unknown, use the properties of triangles to find the remaining angles or sides. Remember that the sum of the angles in a triangle is always 180 degrees. Because of this, if you know two angles, you can easily find the third angle by subtracting the sum of the known angles from 180 degrees.

Step 5: Check for Ambiguous Cases (SSA)

When dealing with the Side-Side-Angle (SSA) case, Check for ambiguous cases — this one isn't optional. So this is because the given information may result in one, two, or no possible triangles. The ambiguous case arises when the given angle is opposite a shorter side compared to the other given side.

The Ambiguous Case (SSA) in Detail

The ambiguous case occurs when we are given two sides and an angle opposite one of those sides (SSA). This situation can result in different possible triangles, depending on the relationship between the given sides and angle. Here are the possible scenarios:

1. No Triangle: If the side opposite the given angle is too short, it will not reach the base, and no triangle can be formed.
2. One Triangle:
   • If the side opposite the given angle is equal to the height of the triangle, a right triangle is formed.
   • If the side opposite the given angle is longer than the adjacent side, only one triangle is possible.
3. Two Triangles: If the side opposite the given angle is shorter than the adjacent side but long enough to reach the base at two different points, two different triangles can be formed.

How to Determine the Number of Possible Triangles

To determine the number of possible triangles in the SSA case, follow these steps:

1. Calculate the Height (h): Calculate the height of the triangle from the given angle to the opposite side. This can be done using the formula:

   h = b * sin(A)

   Where b is the adjacent side, and A is the given angle.

2. Compare the Opposite Side (a) with the Height (h) and the Adjacent Side (b):

   • If a < h, no triangle is possible.
   • If a = h, one triangle (right triangle) is possible.
   • If a > b, one triangle is possible.
   • If h < a < b, two triangles are possible.

Solving for Two Possible Triangles

If the condition h < a < b is met, two triangles are possible. To solve for both triangles:

1. Find the First Angle (B1): Use the Law of Sines to find the first possible value for angle B:

   sin(B1) = (b * sin(A)) / a

   B1 = arcsin((b * sin(A)) / a)

2. Find the Second Angle (B2): The second possible angle B is the supplement of B1:

   B2 = 180° - B1

3. Solve for the Remaining Angles and Sides: For each possible angle B (B1 and B2), solve for the remaining angle C and side c using the properties of triangles and the Law of Sines.

Example Problems

Let's work through some example problems to illustrate the use of the Law of Sines.

Example 1: ASA Case

Given:

• Angle A = 30°
• Angle B = 70°
• Side c = 10 cm

Find:

• Angle C
• Side a
• Side b

Solution:

1. Find Angle C:

   • The sum of angles in a triangle is 180°.
   • C = 180° - A - B = 180° - 30° - 70° = 80°

2. Find Side a:

   • Using the Law of Sines: a / sin(A) = c / sin(C) a / sin(30°) = 10 / sin(80°) a = (10 * sin(30°)) / sin(80°) a ≈ 5.08 cm

3. Find Side b:

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   • Using the Law of Sines: b / sin(B) = c / sin(C) b / sin(70°) = 10 / sin(80°) b = (10 * sin(70°)) / sin(80°) b ≈ 9.54 cm

Because of this, the solution is:

• Angle C = 80°
• Side a ≈ 5.08 cm
• Side b ≈ 9.54 cm

Example 2: AAS Case

Given:

• Angle A = 40°
• Angle B = 60°
• Side a = 8 cm

Find:

• Angle C
• Side b
• Side c

Solution:

1. Find Angle C:

   • C = 180° - A - B = 180° - 40° - 60° = 80°

2. Find Side b:

   • Using the Law of Sines: b / sin(B) = a / sin(A) b / sin(60°) = 8 / sin(40°) b = (8 * sin(60°)) / sin(40°) b ≈ 10.77 cm

3. Find Side c:

   • Using the Law of Sines: c / sin(C) = a / sin(A) c / sin(80°) = 8 / sin(40°) c = (8 * sin(80°)) / sin(40°) c ≈ 12.23 cm

Because of this, the solution is:

• Angle C = 80°
• Side b ≈ 10.77 cm
• Side c ≈ 12.23 cm

Example 3: SSA Case (Ambiguous Case)

Given:

• Angle A = 30°
• Side a = 5 cm
• Side b = 8 cm

Find:

• Angle B
• Angle C
• Side c

Solution:

1. Check for Ambiguous Case:

   • Calculate the height (h): h = b * sin(A) = 8 * sin(30°) = 4 cm
   • Since h < a < b (4 < 5 < 8), there are two possible triangles.

2. Find Angle B1:

   • Using the Law of Sines: sin(B1) = (b * sin(A)) / a = (8 * sin(30°)) / 5 = 0.8 B1 = arcsin(0.8) ≈ 53.13°

3. Find Angle B2:

   • B2 = 180° - B1 = 180° - 53.13° ≈ 126.87°

4. Solve for the First Triangle (Triangle 1):

   • Angle C1 = 180° - A - B1 = 180° - 30° - 53.13° ≈ 96.87°
   • Using the Law of Sines: c1 / sin(C1) = a / sin(A) c1 / sin(96.87°) = 5 / sin(30°) c1 = (5 * sin(96.87°)) / sin(30°) c1 ≈ 9.92 cm

5. Solve for the Second Triangle (Triangle 2):

   • Angle C2 = 180° - A - B2 = 180° - 30° - 126.87° ≈ 23.13°
   • Using the Law of Sines: c2 / sin(C2) = a / sin(A) c2 / sin(23.13°) = 5 / sin(30°) c2 = (5 * sin(23.13°)) / sin(30°) c2 ≈ 3.92 cm

That's why, there are two possible solutions:

• Triangle 1:
  • Angle B1 ≈ 53.13°
  • Angle C1 ≈ 96.87°
  • Side c1 ≈ 9.92 cm
• Triangle 2:
  • Angle B2 ≈ 126.87°
  • Angle C2 ≈ 23.13°
  • Side c2 ≈ 3.92 cm

Common Mistakes and How to Avoid Them

When solving triangles using the Law of Sines, several common mistakes can occur. Here are some of them and how to avoid them:

1. Incorrectly Identifying the Given Information: see to it that you correctly identify the given angles and sides. Double-check the problem statement and label the triangle accordingly.
2. Using the Law of Sines When It's Not Applicable: Remember that the Law of Sines is primarily used in ASA, AAS, and SSA cases. If you have the Side-Angle-Side (SAS) or Side-Side-Side (SSS) case, you should use the Law of Cosines instead.
3. Forgetting to Check for the Ambiguous Case (SSA): Always check for the ambiguous case when you are given two sides and an angle opposite one of them. Determine the number of possible triangles before proceeding with the calculations.
4. Incorrectly Calculating Angles and Sides: Double-check your calculations, especially when using trigonometric functions and inverse trigonometric functions. Make sure your calculator is in the correct mode (degrees or radians).
5. Rounding Errors: Avoid rounding intermediate values, as this can lead to significant errors in the final answer. Keep as many decimal places as possible until the final step, then round to the desired level of precision.
6. Not Verifying the Solution: Once you have found the unknown angles and sides, verify that your solution makes sense. To give you an idea, check that the sum of the angles is 180° and that the longest side is opposite the largest angle.

Applications of the Law of Sines

The Law of Sines has numerous applications in various fields, including:

1. Navigation: Determining distances and angles in navigation, such as calculating the course of a ship or aircraft.
2. Surveying: Measuring distances and angles in land surveying.
3. Engineering: Designing structures and calculating forces in mechanical and civil engineering.
4. Astronomy: Determining distances to stars and planets.
5. Physics: Solving problems involving vectors and forces.

Conclusion

The Law of Sines is a powerful tool for solving triangles when you know certain angles and sides. Consider this: by understanding the conditions for its use, following the steps for solving triangles, and being aware of the ambiguous case, you can accurately determine unknown angles and side lengths. Remember to double-check your work and verify your solution to avoid common mistakes. With practice, you will become proficient in using the Law of Sines to solve a wide range of problems in trigonometry and its applications.