Question

11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)

Explanation:

11)

Step1: Find y via cosine

$\cos(30^\circ)=\frac{y}{3} \implies y=3\times\frac{\sqrt{3}}{2}=\frac{3\sqrt{3}}{2}$

Step2: Find x via sine

$\sin(30^\circ)=\frac{x}{3} \implies x=3\times\frac{1}{2}=\frac{3}{2}$

12)

Step1: Find x via sine

$\sin(30^\circ)=\frac{x}{3} \implies x=3\times\frac{1}{2}=\frac{3}{2}$

Step2: Find other leg via cosine

$\cos(30^\circ)=\frac{\text{adjacent}}{3} \implies \text{adjacent}=3\times\frac{\sqrt{3}}{2}=\frac{3\sqrt{3}}{2}$

13)

Step1: Find y via cosine

$\cos(60^\circ)=\frac{y}{8} \implies y=8\times\frac{1}{2}=4$

Step2: Find x via sine

$\sin(60^\circ)=\frac{x}{8} \implies x=8\times\frac{\sqrt{3}}{2}=4\sqrt{3}$

14)

Step1: Find x via sine

$\sin(60^\circ)=\frac{x}{2\sqrt{3}} \implies x=2\sqrt{3}\times\frac{\sqrt{3}}{2}=3$

Step2: Find other leg via cosine

$\cos(60^\circ)=\frac{\text{adjacent}}{2\sqrt{3}} \implies \text{adjacent}=2\sqrt{3}\times\frac{1}{2}=\sqrt{3}$

15)

Step1: Find b via sine

$\sin(30^\circ)=\frac{b}{4} \implies b=4\times\frac{1}{2}=2$

Step2: Find a via cosine

$\cos(30^\circ)=\frac{a}{4} \implies a=4\times\frac{\sqrt{3}}{2}=2\sqrt{3}$

16)

Step1: Find y via tan

$\tan(60^\circ)=\frac{5\sqrt{3}}{y} \implies y=\frac{5\sqrt{3}}{\sqrt{3}}=5$

Step2: Find x via cosine

$\cos(60^\circ)=\frac{y}{x} \implies x=\frac{5}{\frac{1}{2}}=10$

17)

Step1: Find y via tan

$\tan(30^\circ)=\frac{y}{\sqrt{3}} \implies y=\sqrt{3}\times\frac{1}{\sqrt{3}}=1$

Step2: Find x via cosine

$\cos(30^\circ)=\frac{\sqrt{3}}{x} \implies x=\frac{\sqrt{3}}{\frac{\sqrt{3}}{2}}=2$

18)

Step1: Find y via sin

$\sin(60^\circ)=\frac{y}{\frac{20\sqrt{3}}{3}} \implies y=\frac{20\sqrt{3}}{3}\times\frac{\sqrt{3}}{2}=10$

Step2: Find x via cosine

$\cos(60^\circ)=\frac{x}{\frac{20\sqrt{3}}{3}} \implies x=\frac{20\sqrt{3}}{3}\times\frac{1}{2}=\frac{10\sqrt{3}}{3}$

19)

Step1: Find n via sin

$\sin(30^\circ)=\frac{n}{3} \implies n=3\times\frac{1}{2}=\frac{3}{2}$

Step2: Find m via cosine

$\cos(30^\circ)=\frac{m}{3} \implies m=3\times\frac{\sqrt{3}}{2}=\frac{3\sqrt{3}}{2}$

20)

Step1: Find y via tan

$\tan(60^\circ)=\frac{y}{6\sqrt{3}} \implies y=6\sqrt{3}\times\sqrt{3}=18$

Step2: Find x via cosine

$\cos(60^\circ)=\frac{6\sqrt{3}}{x} \implies x=\frac{6\sqrt{3}}{\frac{1}{2}}=12\sqrt{3}$

21)

Step1: Find y via tan

$\tan(60^\circ)=\frac{y}{4} \implies y=4\times\sqrt{3}=4\sqrt{3}$

Step2: Find x via cosine

$\cos(60^\circ)=\frac{4}{x} \implies x=\frac{4}{\frac{1}{2}}=8$

22)

Step1: Find y via tan

$\tan(30^\circ)=\frac{\frac{\sqrt{2}}{2}}{y} \implies y=\frac{\frac{\sqrt{2}}{2}}{\frac{1}{\sqrt{3}}}=\frac{\sqrt{6}}{2}$

Step2: Find x via cosine

$\cos(30^\circ)=\frac{y}{x} \implies x=\frac{\frac{\sqrt{6}}{2}}{\frac{\sqrt{3}}{2}}=\sqrt{2}$

23)

Step1: Find y via sin

$\sin(30^\circ)=\frac{y}{10} \implies y=10\times\frac{1}{2}=5$

Step2: Find x via cosine

$\cos(30^\circ)=\frac{x}{10} \implies x=10\times\frac{\sqrt{3}}{2}=5\sqrt{3}$

24)

Step1: Find y via cosine

$\cos(30^\circ)=\frac{y}{\frac{\sqrt{6}}{3}} \implies y=\frac{\sqrt{6}}{3}\times\frac{\sqrt{3}}{2}=\frac{\sqrt{2}}{2}$

Step2: Find x via sine

$\sin(30^\circ)=\frac{x}{\frac{\sqrt{6}}{3}} \implies x=\frac{\sqrt{6}}{3}\times\frac{1}{2}=\frac{\sqrt{6}}{6}$

Answer:

1. $x=\frac{3}{2}$, $y=\frac{3\sqrt{3}}{2}$
2. $x=\frac{3}{2}$, adjacent leg $=\frac{3\sqrt{3}}{2}$
3. $x=4\sqrt{3}$, $y=4$
4. $x=3$, adjacent leg $=\sqrt{3}$
5. $a=2\sqrt{3}$, $b=2$
6. $x=10$, $y=5$
7. $x=2$, $y=1$
8. $x=\frac{10\sqrt{3}}{3}$, $y=10$
9. $m=\frac{3\sqrt{3}}{2}$, $n=\frac{3}{2}$
10. $x=12\sqrt{3}$, $y=18$
11. $x=8$, $y=4\sqrt{3}$
12. $x=\sqrt{2}$, $y=\frac{\sqrt{6}}{2}$
13. $x=5\sqrt{3}$, $y=5$
14. $x=\frac{\sqrt{6}}{6}$, $y=\frac{\sqrt{2}}{2}$