Question

which of the following equations could be used to find the value of x?

the figure is not drawn to scale

answer

\\(\frac{\sin(x)}{38} = \frac{\sin(54)}{31}\\)

\\(\cos(x) = \frac{38^2 + 54^2 - 31^2}{2(31)(54)}\\)

\\(x^2 = 38^2 + 31^2 - 2(38)(31)\cos(54)\\)

\\(\cos(x) = \frac{31^2 + 54^2 - 38^2}{2(31)(54)}\\)

Explanation:

Step1: Identify applicable law

We use the Law of Sines, which states that for any triangle, $\frac{\sin(A)}{a} = \frac{\sin(B)}{b}$, where $A,B$ are angles and $a,b$ are the opposite sides.

Step2: Match sides to angles

Angle $x^\circ$ is opposite the side of length 38. Angle $54^\circ$ is opposite the side of length 31. Substitute into the Law of Sines:
$\frac{\sin(x)}{38} = \frac{\sin(54)}{31}$

Step3: Eliminate incorrect options

• The Law of Cosines options are incorrect because we are solving for an angle with two sides and a non-included angle given, not for a side or an angle with three sides.
• The Law of Cosines options have mismatched side-angle pairs for angle $x$.

Answer:

$\frac{\sin(x)}{38} = \frac{\sin(54)}{31}$