Question

which of the following equations could be used to find the value of x?
46°
62
x°
45
the figure is not drawn to scale
answer
$x^2 = 62^2 + 45^2 - 2(62)(45)\cos(46)$ $x^2 = 62^2 + 46^2 - 2(62)(46)\cos(45)$
$\cos(x) = \frac{45^2 + 46^2 - 62^2}{2(45)(46)}$ $\frac{\sin(x)}{62} = \frac{\sin(46)}{45}$

Explanation:

Step1: Recall the Law of Sines

The Law of Sines states that in any triangle, $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$, where $A, B, C$ are the angles and $a, b, c$ are the lengths of the sides opposite those angles respectively.

Step2: Identify the angles and sides

In the given triangle, we have an angle of $46^\circ$ with the side opposite to it being $45$ (wait, no, wait: Wait, the side of length $62$ is adjacent to the $46^\circ$ angle? Wait, no, let's re - identify. Let's label the triangle: Let the angle of $46^\circ$ be angle $A$, the side opposite to angle $A$ is $a = 45$? Wait, no, the side of length $62$: let's see, the angle $x$ is at the vertex with the side of length $45$ opposite? Wait, no, let's do it properly. Let's denote: angle $A=46^\circ$, side $a$ (opposite angle $A$) is the side opposite to $46^\circ$, which is the side of length $45$? Wait, no, the side of length $62$: let's say angle $x$ is angle $B$, side $b$ (opposite angle $B$) is the side of length $62$? Wait, no, the side of length $45$ is opposite to angle $x$? Wait, no, the triangle has sides: one side is $62$, one side is $45$, and the angle between them? Wait, no, the angle of $46^\circ$ is between the side of length $62$ and the other side (let's say the side opposite to angle $x$). Wait, maybe I made a mistake. Let's use the Law of Sines formula. The Law of Sines is $\frac{\sin(\text{angle})}{\text{opposite side}}=\text{constant}$.

Looking at the options, one of the options is $\frac{\sin(x)}{62}=\frac{\sin(46)}{45}$. Let's check: If we consider angle $x$ with opposite side $62$, and angle $46^\circ$ with opposite side $45$, then by Law of Sines, $\frac{\sin(x)}{62}=\frac{\sin(46^\circ)}{45}$.

Now let's check the other options:

• The first option: $x^{2}=62^{2}+45^{2}-2(62)(45)\cos(46)$ is the Law of Cosines. But the Law of Cosines is used when we want to find a side, not an angle. Since we are finding angle $x$, Law of Cosines is not the right approach here (unless we first find the side opposite to $x$ and then use Law of Cosines, but the option is trying to find $x^{2}$ which is for a side, not an angle).

• The second option: $x^{2}=62^{2}+46^{2}-2(62)(46)\cos(45)$: 46 is an angle, not a side length, so this is incorrect.

• The third option: $\cos(x)=\frac{45^{2}+46^{2}-62^{2}}{2(45)(46)}$: 46 is an angle, not a side length, so this is incorrect.

• The fourth option: $\frac{\sin(x)}{62}=\frac{\sin(46)}{45}$: This follows the Law of Sines, where angle $x$ has opposite side $62$, and angle $46^\circ$ has opposite side $45$. So this equation can be used to find $x$.

Answer:

$\boldsymbol{\frac{\sin(x)}{62}=\frac{\sin(46)}{45}}$