Question

1. determine side c .
2. in an obtuse triangle the measurement of two angles are 105.00 degrees and 19.00 degrees. the measurement of the side opposite angle 105 degrees is 20.700 centimeters and opposite angle 19.00 degrees is c.

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 known values

We know that angle $A = 105.00^\circ$ with opposite side $a = 20.700$ cm, angle $B = 19.00^\circ$ with opposite side $b = c$ (we need to find $c$).

Step3: Apply the Law of Sines

Using the Law of Sines, we have $\frac{c}{\sin 19.00^\circ} = \frac{20.700}{\sin 105.00^\circ}$.

First, calculate $\sin 105.00^\circ$ and $\sin 19.00^\circ$.
$\sin 105^\circ = \sin(60^\circ + 45^\circ) = \sin 60^\circ \cos 45^\circ + \cos 60^\circ \sin 45^\circ = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{6} + \sqrt{2}}{4} \approx 0.9659$.
$\sin 19.00^\circ \approx 0.3256$.

Now, solve for $c$:
$c = \frac{20.700 \cdot \sin 19.00^\circ}{\sin 105.00^\circ}$
Substitute the values:
$c = \frac{20.700 \cdot 0.3256}{0.9659}$
First, calculate the numerator: $20.700 \cdot 0.3256 \approx 20.700 \times 0.3256 = 6.74$.
Then divide by the denominator: $c \approx \frac{6.74}{0.9659} \approx 6.98$ (rounded to two decimal places).

Answer:

The length of side \( c \) is approximately \( 6.98 \) centimeters.