Find  
c
  if  
a
=
2.27
  miles,  
b
=
3.1
  miles and  
∠
C
=
40.5
  degrees.
(Round the answer to the nearest hundredth.)
c
=
  
   mi


Assume  
∠
A
  is opposite side  
a
 ,  
∠
B
  is opposite side  
b
 , and  
∠
C
  is opposite side  
c
 .


To solve for \( c \), we can apply the **Law of Cosines**. The Law of Cosines is particularly useful when we know two sides of a triangle and the angle between them (in this case, sides \( a \) and \( b \), and the angle \( \angle C \)).

The formula for the Law of Cosines is:

\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]

Given:
- \( a = 2.27 \) miles
- \( b = 3.1 \) miles
- \( \angle C = 40.5^\circ \)

First, convert the angle \( \angle C \) to radians since trigonometric functions typically use radians. However, since we have the angle in degrees, we can directly use the cosine of the degree value.

Now, applying the values to the Law of Cosines formula:

\[
c^2 = (2.27)^2 + (3.1)^2 - 2(2.27)(3.1) \cdot \cos(40.5^\circ)
\]

Let's calculate this step by step.

1. \( 2.27^2 = 5.1529 \)
2. \( 3.1^2 = 9.61 \)
3. \( \cos(40.5^\circ) \approx 0.7604 \)
4. \( 2(2.27)(3.1) \cdot 0.7604 \approx 10.6937 \)

Now, substitute into the equation:

\[
c^2 = 5.1529 + 9.61 - 10.6937
\]

\[
c^2 = 15.6109 - 10.6937 = 4.9172
\]

Finally, take the square root of both sides to find \( c \):

\[
c = \sqrt{4.9172} \approx 2.22 \, \text{miles}
\]

Thus, \( c \approx 2.22 \) miles (rounded to the nearest hundredth).

---

Would you like more details or have any questions?

Here are 5 related questions:
1. How does the Law of Cosines relate to the Pythagorean theorem?
2. Can we use the Law of Sines in this problem?
3. What is the cosine of an obtuse angle, and how does it affect the Law of Cosines?
4. If angle \( C \) was larger, how would that affect side \( c \)?
5. What are some real-world applications of the Law of Cosines?

**Tip**: The Law of Cosines can be very useful when you have an angle other than \( 90^\circ \), unlike the Pythagorean theorem which only applies to right-angled triangles!

Find c if a = 2.27 miles, b = 3.1 miles and ∠C = 40.5 degrees. (Round the answer to the nearest hundredth.) Assume ∠A is opposite side a, ∠B is opposite side b, and ∠C is opposite side c.

Learn how to find the missing side of a triangle using the Law of Cosines. Given sides a = 2.27 miles, b = 3.1 miles, and angle C = 40.5 degrees, this step-by-step solution explains how to apply the Law of Cosines to calculate side c, rounded to the nearest hundredth. Ideal for students in Grades 10-12.

Math Problem Statement

Find
c if
a

2.27 miles,
b

3.1 miles and
∠ C

40.5 degrees. (Round the answer to the nearest hundredth.) c

Solution

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