Question

b.
hint (b):
use the
law of
sines.

c.

hint (c):
use the
law of
cosines.
answer (c):
$x$
$\approx 15.50^{\circ}$

Explanation:

Step1: Find third angle of triangle

Sum of angles in triangle is $180^\circ$.
$\text{Third angle} = 180^\circ - 70^\circ - 23^\circ = 87^\circ$

Step2: Apply Law of Sines

Relate sides and opposite angles.
$\frac{x}{\sin(70^\circ)} = \frac{10}{\sin(23^\circ)}$

Step3: Solve for $x$

Rearrange and calculate the value.
$x = \frac{10 \times \sin(70^\circ)}{\sin(23^\circ)} \approx \frac{10 \times 0.9397}{0.3907} \approx 24.05$

Step4: Apply Law of Cosines for part (c)

Use the formula for angle: $\cos(x) = \frac{a^2 + b^2 - c^2}{2ab}$
$\cos(x) = \frac{10^2 + 11^2 - 3^2}{2 \times 10 \times 11} = \frac{100 + 121 - 9}{220} = \frac{212}{220} \approx 0.9636$

Step5: Calculate angle $x$

Take inverse cosine of the value.
$x = \cos^{-1}(0.9636) \approx 15.50^\circ$

Answer:

For part b: $x \approx 24.05$
For part c: $x \approx 15.50^\circ$