Question

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

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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: Identify triangle properties

We have a triangle with: angle $46^\circ$, side opposite this angle = $45$; angle $x^\circ$, side opposite this angle = $62$.

Step2: Recall the Law of Sines

The Law of Sines states $\frac{\sin(A)}{a} = \frac{\sin(B)}{b}$, where $A,B$ are angles, $a,b$ are their opposite sides.

Step3: Match values to the formula

Substitute $A=x$, $a=62$, $B=46$, $b=45$ into the Law of Sines:
$\frac{\sin(x)}{62} = \frac{\sin(46)}{45}$

Step4: Eliminate incorrect options

• Option 1: Uses Law of Cosines for side $x$, but $x$ is an angle, not a side.
• Option 2: Incorrect side-angle pairing in Law of Cosines.
• Option 3: Incorrect side-angle pairing in Law of Cosines for angle $x$.

Answer:

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