Understanding Law of Cosines Word Problems: A Comprehensive Guide
Law of cosines word problems are common in trigonometry, especially when dealing with non-right triangles. These problems often involve finding missing sides or angles in triangles where traditional right triangle methods like the Pythagorean theorem do not apply. Mastering how to approach and solve these word problems is essential for students and professionals alike, as they appear frequently in geometry, engineering, navigation, and physics. This article aims to provide a detailed understanding of the law of cosines, how to interpret word problems involving it, and step-by-step strategies to find solutions efficiently.
What Is the Law of Cosines?
The law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is especially useful in oblique (non-right) triangles.
Mathematical Formula
Given a triangle with sides \(a\), \(b\), and \(c\), and corresponding opposite angles \(A\), \(B\), and \(C\):
\[
c^2 = a^2 + b^2 - 2ab \cos C
\]
Similarly,
\[
a^2 = b^2 + c^2 - 2bc \cos A
\]
\[
b^2 = a^2 + c^2 - 2ac \cos B
\]
These formulas allow you to find an unknown side or angle when enough information is provided.
When to Use the Law of Cosines
Use the law of cosines in the following scenarios:
- When you know two sides and the included angle (SAS) and want to find the third side.
- When you know all three sides (SSS) and want to find an angle.
- When dealing with non-right triangles where the Pythagorean theorem cannot be applied directly.
Key Steps in Solving Law of Cosines Word Problems
Approaching law of cosines word problems systematically makes the process less intimidating. Here’s a step-by-step guide:
1. Read and Understand the Problem Carefully
Identify what is given:
- Sides lengths
- Angles
- The unknown quantity (side or angle)
Determine what you need to find.
2. Draw a Clear Diagram
Visual representation helps in understanding the problem layout:
- Sketch the triangle with labeled sides and angles.
- Mark known and unknown quantities clearly.
3. Decide Which Law of Cosines Formula to Use
Based on what information is provided:
- If two sides and the included angle are known, use the SAS form.
- If all three sides are known, use the SSS form to find an angle.
4. Substitute Known Values and Solve
- Plug the known values into the formula.
- Rearrange the formula algebraically to isolate the unknown.
- Use a calculator to compute the result, ensuring correct mode (degrees or radians).
5. Check the Reasonableness of Your Answer
- Verify that the sides and angles satisfy triangle inequalities.
- Confirm that angles are within 0° to 180°.
- Make sense of the solution in the context of the problem.
Examples of Law of Cosines Word Problems
To solidify understanding, let’s explore several example problems with detailed solutions.
Example 1: Finding a Side in a Triangle (SAS Case)
Problem:
A triangle has sides \(a = 8\, \text{units}\), \(b = 6\, \text{units}\), and the included angle \(C = 60^\circ\). Find side \(c\).
Solution:
Step 1: Write the law of cosines formula:
\[
c^2 = a^2 + b^2 - 2ab \cos C
\]
Step 2: Substitute known values:
\[
c^2 = 8^2 + 6^2 - 2 \times 8 \times 6 \times \cos 60^\circ
\]
\[
c^2 = 64 + 36 - 2 \times 8 \times 6 \times 0.5
\]
\[
c^2 = 100 - 2 \times 8 \times 6 \times 0.5
\]
\[
c^2 = 100 - (2 \times 8 \times 6 \times 0.5)
\]
Calculate the second term:
\[
2 \times 8 \times 6 \times 0.5 = 2 \times 8 \times 6 \times 0.5 = (2 \times 8) \times 6 \times 0.5 = 16 \times 6 \times 0.5 = 96 \times 0.5 = 48
\]
So:
\[
c^2 = 100 - 48 = 52
\]
Step 3: Find \(c\):
\[
c = \sqrt{52} \approx 7.21\, \text{units}
\]
Answer: The length of side \(c\) is approximately 7.21 units.
---
Example 2: Finding an Angle in a Triangle (SSS Case)
Problem:
Sides of a triangle are \(a = 7\, \text{units}\), \(b = 9\, \text{units}\), and \(c = 10\, \text{units}\). Find angle \(C\).
Solution:
Step 1: Use the law of cosines formula for angle \(C\):
\[
c^2 = a^2 + b^2 - 2ab \cos C
\]
Rearranged for \(\cos C\):
\[
\cos C = \frac{a^2 + b^2 - c^2}{2ab}
\]
Step 2: Substitute known values:
\[
\cos C = \frac{7^2 + 9^2 - 10^2}{2 \times 7 \times 9}
\]
\[
\cos C = \frac{49 + 81 - 100}{2 \times 7 \times 9}
\]
\[
\cos C = \frac{130 - 100}{126} = \frac{30}{126} = \frac{5}{21} \approx 0.2381
\]
Step 3: Find \(C\):
\[
C = \cos^{-1}(0.2381) \approx 76.2^\circ
\]
Answer: Angle \(C\) measures approximately 76.2°.
---
Example 3: Application in Real-Life Context
Problem:
A boat is anchored at point A on the shore. From point A, the boat is 300 meters east of a lighthouse (point B). The boat then moves 400 meters north, reaching point C. What is the straight-line distance from the lighthouse to the boat’s new position?
Solution:
Step 1: Visualize and Label:
- Point B (lighthouse)
- Point A (initial position of boat)
- Point C (new position of boat)
Coordinates:
- \(A = (0, 0)\)
- \(B = (300, 0)\)
- \(C = (300, 400)\)
The problem asks for the distance between B and C, i.e., the length of side \(BC\).
Step 2: Calculate side \(BC\):
- Coordinates of B: (300, 0)
- Coordinates of C: (300, 400)
Distance \(BC\):
\[
BC = \sqrt{(300 - 300)^2 + (400 - 0)^2} = \sqrt{0 + 160,000} = 400\, \text{meters}
\]
Note: Since the points are aligned vertically, the distance is simply 400 meters.
Alternative scenario: If the problem asks for the distance from the lighthouse to the boat’s position after moving, and the movement is at an angle, then the law of cosines becomes relevant.
---
Common Mistakes to Avoid in Law of Cosines Word Problems
- Mixing degrees and radians: Always ensure your calculator is in the correct mode.
- Incorrectly identifying the known parts: Clarify whether you have SAS or SSS information.
- Forgetting to verify the triangle inequality: The sum of two sides must be greater than the third.
- Mislabeling sides and angles: Keep consistent with notation throughout calculations.
- Neglecting to convert angles to radians when necessary: Usually, for cosine calculations, degrees are fine, but check your calculator settings.
Practice Problems to Enhance Your Skills
1. A triangle has sides of 5, 7, and 10 units. Find the measure of the angle opposite the side of length 7 units.
2. In triangle ABC, side \(a = 8\) units, side \(b = 6\) units, and angle \(A = 45^\circ\). Find side \(c\).
3. Two towers are 150 meters apart. From the top of one tower, the angle of elevation to the top of the other tower is
Frequently Asked Questions
How do you set up a law of cosines problem when given two sides and the included angle?
Use the formula c² = a² + b² - 2ab cos(C), where a and b are the given sides and C is the included angle. Plug in the known values to find the unknown side c.
What is the first step to solve a triangle using the law of cosines when given three sides?
Identify the side opposite the largest angle and apply the law of cosines formula to find an angle, then use the law of sines or cosines to find the remaining angles and sides.
How can the law of cosines help in finding the missing side in a non-right triangle?
By applying the law of cosines with the known sides and included angle, you can solve for the unknown side without needing a right angle or altitude.
When solving a word problem involving distances between points, how do you determine which law of cosines formula to use?
Identify if you know two sides and the included angle or all three sides. Use the law of cosines accordingly: for side length calculations, when given two sides and an included angle, or for angles when all sides are known.
How do you apply the law of cosines to find the angle between two known sides?
Rearrange the law of cosines formula to solve for the angle: cos(C) = (a² + b² - c²) / (2ab). Plug in the known sides to find the cosine of the angle, then take the inverse cosine.
What are common mistakes to avoid when solving law of cosines word problems?
Common mistakes include mixing up sides and angles, using the wrong formula (law of sines vs. law of cosines), forgetting to convert angles to radians if necessary, and miscalculating the square or square root operations.
Can the law of cosines be used to solve for an angle in a triangle with two known sides and an opposite side?
Yes, if you know two sides and the included angle, you can use the law of cosines to find the third side. Conversely, if you know all three sides, you can find any angle by rearranging the formula accordingly.
