Answer
The measurement of side ‘a’ is \[43\text{ cm}\], measurement of side ‘c’ is\[33\text{ cm}\] and the measurement of angle B is \[38{}^\circ \].
Work Step by Step
Substitute the values in the above function as follows:
\[\begin{align}
& \sin \text{52}{}^\circ =\frac{a}{54} \\
& a=\sin \text{52}{}^\circ \times 54 \\
& =0.788\times 54 \\
& \approx 43
\end{align}\]
It implies that: \[a\approx 43\text{ cm}\]
Now, in order to compute the value of c, use the trigonometric ratio of cos A as follows:
\[\cos A=\frac{\text{Side adjacent to angle }A}{\text{Hypotenuse}}\]
Substitute the values in the above function as follows:
\[\begin{align}
& \cos \text{52}{}^\circ =\frac{b}{54} \\
& b=\cos \text{52}{}^\circ \times 54 \\
& =0.615\times 54 \\
& \approx 33
\end{align}\]
It implies that: \[b\approx 33\text{ cm}\]
Now, compute the third missing angle using angle sum property which specifies that sum of all angles of a triangle is equal to 180 degree.
Accordingly,
\[\begin{align}
& m\measuredangle A+m\measuredangle B+m\measuredangle C=180{}^\circ \\
& 52{}^\circ +m\measuredangle B+90{}^\circ =180{}^\circ \\
& m\measuredangle B=180{}^\circ -90{}^\circ -52{}^\circ \\
& =38{}^\circ
\end{align}\]
It implies that: \[\measuredangle B=38{}^\circ \]
Hence, the measurement of side denoted by ‘a’ is \[43\text{ cm}\], measurement of side denoted by ‘b’ is\[33\text{ cm}\] and the measurement of angle B is \[38{}^\circ \].
