Question

which of the following equations could be used to find the value of x?
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answer
$x^2 = 37^2 + 56^2 - 2(37)(56)\cos(69)$
$\cos(x) = \frac{56^2+69^2-37^2}{2(56)(69)}$
$x^2 = 37^2 + 69^2 - 2(37)(69)\cos(56)$
$\frac{\sin(x)}{37} = \frac{\sin(69)}{56}$

Explanation:

Step1: Identify applicable law

We use the Law of Sines, which states that for any triangle, the ratio of the sine of an angle to the length of its opposite side is constant: $\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}$.

Step2: Match angles and sides

Angle $x^\circ$ is opposite the side of length 37, and angle $69^\circ$ is opposite the side of length 56. Substitute these into the Law of Sines:
$\frac{\sin(x)}{37} = \frac{\sin(69)}{56}$

Step3: Eliminate incorrect options

• The first option uses the Law of Cosines to find the side opposite $69^\circ$, not angle $x$.
• The second option misassigns values in the Law of Cosines.
• The third option misapplies the Law of Cosines for angle $x$ (uses incorrect side values).

Answer:

$\frac{\sin(x)}{37} = \frac{\sin(69)}{56}$