Question

1. determine side x. 13. in a scalene triangle the measurement of two angles are 81.35 degrees and 83.60 degrees. the measurement of the side opposite angle 93.60 degrees is x and opposite angle 81.35 degrees is 51.080 millimeters.

Explanation:

Step1: Recall the Law of Sines

The Law of Sines states that in any triangle, $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$, where $a$, $b$, $c$ are the lengths of the sides opposite angles $A$, $B$, $C$ respectively.

Step2: Identify the given values

We know two angles: $A = 93.60^\circ$ (opposite side $x$), $B = 81.35^\circ$ (opposite side $b = 51.080$ mm), and we need to find $x$.

Step3: Apply the Law of Sines

Using the Law of Sines, we have $\frac{x}{\sin A} = \frac{b}{\sin B}$. Substituting the known values: $\frac{x}{\sin(93.60^\circ)} = \frac{51.080}{\sin(81.35^\circ)}$.

Step4: Solve for $x$

First, calculate $\sin(93.60^\circ)$ and $\sin(81.35^\circ)$.

• $\sin(93.60^\circ) \approx \sin(90^\circ + 3.60^\circ) = \cos(3.60^\circ) \approx 0.998$ (using the co - function identity $\sin(90^\circ+\theta)=\cos\theta$)
• $\sin(81.35^\circ) \approx 0.990$

Then, $x=\frac{51.080\times\sin(93.60^\circ)}{\sin(81.35^\circ)}$. Substituting the approximate values of sines: $x=\frac{51.080\times0.998}{0.990}$.

First, calculate the numerator: $51.080\times0.998 = 51.080\times(1 - 0.002)=51.080-0.10216 = 50.97784$

Then, divide by the denominator: $x=\frac{50.97784}{0.990}\approx51.49$ mm (more accurately, using a calculator for the sines:
$\sin(93.60^\circ)=\sin(93.60)=\sin(90 + 3.6)=\cos(3.6)\approx\cos(3.6\times\frac{\pi}{180})\approx\cos(0.0628)\approx0.9980$
$\sin(81.35^\circ)=\sin(81.35\times\frac{\pi}{180})\approx\sin(1.420)\approx0.9903$

So $x=\frac{51.080\times\sin(93.60^\circ)}{\sin(81.35^\circ)}=\frac{51.080\times0.9980}{0.9903}=\frac{51.080\times0.9980}{0.9903}\approx\frac{50.97784}{0.9903}\approx51.48$ mm)

Answer:

The length of side $x$ is approximately $\boldsymbol{51.48}$ mm (or depending on the precision of the sine values, it can be calculated more accurately. If we use a calculator for more precise sine values:

$\sin(93.60^\circ)= \sin(93.6)=\sin(93 + 0.6)=\sin(93^\circ36')\approx0.99803$

$\sin(81.35^\circ)=\sin(81^\circ21')\approx0.99034$

Then $x=\frac{51.080\times0.99803}{0.99034}=\frac{51.080\times0.99803}{0.99034}\approx\frac{50.977}{0.99034}\approx51.47$ mm)