Which equation correctly uses the law of cosines to solve for the length [tex]$s$[/tex]?

[tex]$9^2=s^2+10^2-2(s)(10) \cos \left(100^{\circ}\right)$[/tex]
[tex]$9=s+10-2(s)(10) \cos \left(100^{\circ}\right)$[/tex]
[tex]$10^2=s^2+100-2(s)(10) \cos \left(100^{\circ}\right)$[/tex]
[tex]$s^2=9^2+10^2-2(9)(10) \cos \left(100^{\circ}\right)$[/tex]

The correct equation that uses the Law of Cosines to solve for the length s is s 2 = 9 2 + 1 0 2 − 2 ( 9 ) ( 10 ) cos ( 10 0 \textcirc ) . This equation accurately applies the Law of Cosines, placing s opposite the angle of 10 0 \textcirc . Therefore, the answer is s 2 = 9 2 + 1 0 2 − 2 ( 9 ) ( 10 ) cos ( 10 0 \textcirc ) .
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Answered by Anonymous | 2025-07-04

The Law of Cosines relates the sides and angles of a triangle: c 2 = a 2 + b 2 − 2 ab cos ( C ) .
Analyze each equation to see if it correctly applies the Law of Cosines.
Identify the equation where s is isolated, making it easier to solve for s .
Conclude that the correct equation is: s 2 = 9 2 + 1 0 2 − 2 ( 9 ) ( 10 ) cos ( 10 0 ∘ ) ​ .

Explanation

Analyze the Law of Cosines Let's analyze the given equations and compare them with the Law of Cosines to identify the correct one. The Law of Cosines states that for any triangle with sides of lengths a , b , and c , and an angle C opposite the side of length c , the following equation holds:

c 2 = a 2 + b 2 − 2 ab cos ( C )
We need to find the equation that correctly applies this law to solve for the length s .

Examine each equation Now, let's examine each of the given equations:

9 2 = s 2 + 1 0 2 − 2 ( s ) ( 10 ) cos ( 10 0 ∘ )

This equation implies that the side of length 9 is opposite the angle of 100 degrees, and the other two sides have lengths s and 10. This is a valid application of the Law of Cosines.


9 = s + 10 − 2 ( s ) ( 10 ) cos ( 10 0 ∘ )

This equation is not in the form of the Law of Cosines. It seems to be an incorrect attempt to relate the side lengths and the cosine of the angle.


1 0 2 = s 2 + 100 − 2 ( s ) ( 10 ) cos ( 10 0 ∘ )

This equation can be rewritten as 1 0 2 = s 2 + 1 0 2 − 2 ( s ) ( 10 ) cos ( 10 0 ∘ ) . This implies that the side of length 10 is opposite the angle of 100 degrees, and the other two sides have lengths s and 10. However, this simplifies to 0 = s 2 − 20 s cos ( 10 0 ∘ ) , which means s must be 0 or 20 cos ( 10 0 ∘ ) . This is a special case and might not be generally correct.


s 2 = 9 2 + 1 0 2 − 2 ( 9 ) ( 10 ) cos ( 10 0 ∘ )

This equation implies that the side of length s is opposite the angle of 100 degrees, and the other two sides have lengths 9 and 10. This is also a valid application of the Law of Cosines.


Identify the correct equation Comparing equations 1 and 4, we see that both are in the correct form of the law of cosines. However, the question asks to solve for the length s . Equation 4 directly gives s 2 on one side, making it easier to solve for s . Therefore, equation 4 is the correct one.

State the final answer The correct equation that uses the Law of Cosines to solve for the length s is:


s 2 = 9 2 + 1 0 2 − 2 ( 9 ) ( 10 ) cos ( 10 0 ∘ )
Examples
The Law of Cosines is incredibly useful in fields like surveying and navigation. Imagine you're a surveyor trying to determine the distance across a lake. You can measure the distances from your position to two points on opposite sides of the lake, as well as the angle between those lines of sight. Using the Law of Cosines, you can then calculate the distance across the lake without needing to physically cross it! This principle is also used in GPS technology to calculate distances between satellites and receivers, helping you navigate your way around town.

Answered by GinnyAnswer | 2025-07-04