Question

in δdef, d = 470 inches, f = 360 inches and ∠f = 146°. find all possible values of ∠d, to the nearest degree. answer attempt 2 out of 2 additional angle no possible angles submit answer

Explanation:

Step1: Apply the Law of Sines

By the Law of Sines, $\frac{\sin D}{d}=\frac{\sin F}{f}$. Substituting the given values $d = 470$, $f = 360$, and $\angle F=146^{\circ}$, we get $\sin D=\frac{d\sin F}{f}=\frac{470\times\sin(146^{\circ})}{360}$.

Step2: Calculate the value of $\sin D$

First, $\sin(146^{\circ})\approx0.5592$. Then $\sin D=\frac{470\times0.5592}{360}=\frac{262.824}{360}\approx0.7301$.

Step3: Find the principal - value of $\angle D$

Using the inverse - sine function, $D_1=\sin^{-1}(0.7301)\approx47^{\circ}$.

Step4: Find the supplementary angle

Since the sine function is positive in the second quadrant, the supplementary angle $D_2 = 180^{\circ}-47^{\circ}=133^{\circ}$. But we need to check if this is valid. In a triangle, the sum of all angles must be $180^{\circ}$. If $F = 146^{\circ}$ and $D_2=133^{\circ}$, then $F + D_2>180^{\circ}$. So, the only valid value of $\angle D$ is the principal - value.

Answer:

$47^{\circ}$