find the measure of the acute angle, $x$, in degrees. the figure is drawn to scale. round to the hundredths…
find the measure of the acute angle, $x$, in degrees. the figure is drawn to scale. round to the hundredths place.
Answer
Explanation:
Step1: Identify the Law of Sines
The Law of Sines relates side lengths to opposite angles. $$ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} $$
Step2: Substitute the given values
Substitute the sides $16$ and $20$ and their opposite angles $x$ and $49^{\circ}$. $$ \frac{16}{\sin(49^{\circ})} = \frac{20}{\sin(x)} $$
Step3: Solve for $\sin(x)$
Rearrange the equation to isolate $\sin(x)$. $$ \sin(x) = \frac{20 \cdot \sin(49^{\circ})}{16} $$
Step4: Calculate the value of $\sin(x)$
Compute the numerical value using $\sin(49^{\circ}) \approx 0.7547$. $$ \sin(x) \approx \frac{20 \cdot 0.7547}{16} \approx 0.943387 $$
Step5: Find the angle $x$
Apply the inverse sine function to find the measure of $x$. $$ x = \arcsin(0.943387) \approx 70.626^{\circ} $$
Step6: Round to the hundredths place
Round the final result to two decimal places. $$ x \approx 70.63^{\circ} $$
Answer:
70.63^{\circ}