Question

1 solve for the unknown in each triangle. round each answer to the nearest tenth.
a
triangle with side ac = 7 m, side bc = 5 m, angle b = 53°, unknown angle a = $x^circ$
b
triangle with side length 9 in, side length 12 in, included angle = 102°, unknown side = $x$
c
triangle with side length 11 ft, angle = 118°, angle = 29°, unknown side = $x$
d
triangle with side length 5.9 cm, side length 3.1 cm, side length 4.3 cm, unknown angle = $x^circ$

Explanation:

Part (a)

Step1: Apply Law of Sines

$\frac{\sin x^\circ}{5} = \frac{\sin 53^\circ}{7}$

Step2: Isolate $\sin x^\circ$

$\sin x^\circ = \frac{5 \sin 53^\circ}{7}$

Step3: Calculate $\sin x^\circ$ value

$\sin x^\circ = \frac{5 \times 0.7986}{7} \approx 0.5704$

Step4: Solve for $x$

$x = \arcsin(0.5704) \approx 34.8^\circ$

Part (b)

Step1: Apply Law of Cosines

$x^2 = 9^2 + 12^2 - 2(9)(12)\cos 102^\circ$

Step2: Compute squared terms

$x^2 = 81 + 144 - 216(-0.2079)$

Step3: Simplify right-hand side

$x^2 = 225 + 44.9064 = 269.9064$

Step4: Solve for $x$

$x = \sqrt{269.9064} \approx 16.4$ in

Part (c)

Step1: Find third angle

Third angle $= 180^\circ - 118^\circ - 29^\circ = 33^\circ$

Step2: Apply Law of Sines

$\frac{x}{\sin 118^\circ} = \frac{11}{\sin 33^\circ}$

Step3: Isolate $x$

$x = \frac{11 \sin 118^\circ}{\sin 33^\circ}$

Step4: Calculate $x$ value

$x = \frac{11 \times 0.8829}{0.5446} \approx 17.9$ ft

Part (d)

Step1: Apply Law of Cosines

$\cos x^\circ = \frac{4.3^2 + 5.9^2 - 3.1^2}{2(4.3)(5.9)}$

Step2: Compute squared terms

$\cos x^\circ = \frac{18.49 + 34.81 - 9.61}{50.54}$

Step3: Simplify right-hand side

$\cos x^\circ = \frac{43.69}{50.54} \approx 0.8645$

Step4: Solve for $x$

$x = \arccos(0.8645) \approx 30.2^\circ$

Answer:

a. $34.8^\circ$
b. $16.4$ in
c. $17.9$ ft
d. $30.2^\circ$