Question

law of sines
9)
10)
law of cosines
11)
12)

Explanation:

Problem 9

Step1: Identify opposite angles/sides

Angle opposite $x$: $34^\circ$, angle opposite side 5: $27^\circ$

Step2: Apply Law of Sines

$$\frac{x}{\sin(34^\circ)} = \frac{5}{\sin(27^\circ)}$$

Step3: Solve for $x$

$$x = \frac{5\sin(34^\circ)}{\sin(27^\circ)}$$
Calculate $\sin(34^\circ)\approx0.5592$, $\sin(27^\circ)\approx0.4540$
$$x\approx\frac{5\times0.5592}{0.4540}\approx6.15$$

Problem 10

Step1: Identify opposite angles/sides

Angle opposite $x$: side 17, angle opposite side 35: $92^\circ$

Step2: Apply Law of Sines

$$\frac{\sin(x^\circ)}{17} = \frac{\sin(92^\circ)}{35}$$

Step3: Solve for $x$

$$\sin(x^\circ) = \frac{17\sin(92^\circ)}{35}$$
Calculate $\sin(92^\circ)\approx0.9994$
$$\sin(x^\circ)\approx\frac{17\times0.9994}{35}\approx0.4854$$
$$x^\circ\approx\arcsin(0.4854)\approx29.0^\circ$$

Problem 11

Step1: Apply Law of Cosines

$$x^2 = 22^2 + 27^2 - 2(22)(27)\cos(73^\circ)$$

Step2: Calculate each term

$22^2=484$, $27^2=729$, $2(22)(27)=1188$, $\cos(73^\circ)\approx0.2924$
$$x^2=484+729-1188\times0.2924$$
$$x^2=1213-347.37$$
$$x^2\approx865.63$$

Step3: Solve for $x$

$$x\approx\sqrt{865.63}\approx29.42$$

Problem 12

Step1: Apply Law of Cosines

$$\cos(x^\circ) = \frac{10^2 + 17^2 - 11^2}{2(10)(17)}$$

Step2: Calculate each term

$10^2=100$, $17^2=289$, $11^2=121$, $2(10)(17)=340$
$$\cos(x^\circ)=\frac{100+289-121}{340}=\frac{268}{340}\approx0.7882$$

Step3: Solve for $x$

$$x^\circ\approx\arccos(0.7882)\approx37.0^\circ$$

Answer:

1. $x\approx6.15$
2. $x\approx29.0^\circ$
3. $x\approx29.42$
4. $x\approx37.0^\circ$