Question

Question was provided via image upload.

Explanation:

Step1: Calculate cos(72°) to 4 decimals

$\cos(72^\circ) \approx 0.3090$

Step2: Calculate sin(35°) to 4 decimals

$\sin(35^\circ) \approx 0.5736$

Step3: Calculate tan(61°) to 4 decimals

$\tan(61^\circ) \approx 1.8040$

Step4: Calculate sin(30°) to 4 decimals

$\sin(30^\circ) = 0.5000$

Step5: Calculate cos(30°) to 4 decimals

$\cos(30^\circ) \approx 0.8660$

Step6: Calculate tan(30°) to 4 decimals

$\tan(30^\circ) \approx 0.5774$

Step7: Find sin A for triangle ABC

$\sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{30}{34} = \frac{15}{17}$

Step8: Find tan B for triangle ABC

$\tan B = \frac{\text{opposite}}{\text{adjacent}} = \frac{16}{30} = \frac{8}{15}$

Step9: Find cos A for triangle ABC

$\cos A = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{16}{34} = \frac{8}{17}$

Step10: Find cos B for triangle ABC

$\cos B = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{30}{34} = \frac{15}{17}$

Step11: Find sin D for triangle DEF

$\sin D = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{16}{20} = \frac{4}{5}$

Step12: Find tan E for triangle DEF

$\tan E = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{16} = \frac{3}{4}$

Step13: Find cos E for triangle DEF

$\cos E = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{16}{20} = \frac{4}{5}$

Step14: Find cos D for triangle DEF

$\cos D = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{20} = \frac{3}{5}$

Step15: Solve for y in $\sin64^\circ=\frac{y}{18}$

$y = 18 \times \sin(64^\circ) \approx 18 \times 0.8988 = 16.18$

Step16: Solve for x in $\cos64^\circ=\frac{x}{18}$

$x = 18 \times \cos(64^\circ) \approx 18 \times 0.4384 = 7.89$

Step17: Solve for y in $\cos38^\circ=\frac{14}{y}$

$y = \frac{14}{\cos(38^\circ)} \approx \frac{14}{0.7880} = 17.77$

Step18: Solve for x in $\tan38^\circ=\frac{x}{14}$

$x = 14 \times \tan(38^\circ) \approx 14 \times 0.7813 = 10.94$

Step19: Solve for y in $\cos19^\circ=\frac{15}{y}$

$y = \frac{15}{\cos(19^\circ)} \approx \frac{15}{0.9455} = 15.86$

Step20: Solve for x in $\tan19^\circ=\frac{x}{15}$

$x = 15 \times \tan(19^\circ) \approx 15 \times 0.3443 = 5.16$

Step21: Solve for x in $\sin52^\circ=\frac{x}{15}$

$x = 15 \times \sin(52^\circ) \approx 15 \times 0.7880 = 11.82$

Step22: Solve for y in $\cos52^\circ=\frac{y}{15}$

$y = 15 \times \cos(52^\circ) \approx 15 \times 0.6157 = 9.24$

Step23: Solve for y in $\sin37^\circ=\frac{12}{y}$

$y = \frac{12}{\sin(37^\circ)} \approx \frac{12}{0.6018} = 19.94$

Step24: Solve x,y for triangle 1

$x = 5 \times \tan(49^\circ) \approx 5 \times 1.1504 = 5.75$
$y = \frac{5}{\cos(49^\circ)} \approx \frac{5}{0.6561} = 7.62$

Step25: Solve x,y for triangle 2

$x = 9 \times \sin(11^\circ) \approx 9 \times 0.1908 = 1.72$
$y = 9 \times \cos(11^\circ) \approx 9 \times 0.9816 = 8.83$

Step26: Solve x,y for triangle 3

$x = 2 \times \sin(64^\circ) \approx 2 \times 0.8988 = 1.80$
$y = 2 \times \cos(64^\circ) \approx 2 \times 0.4384 = 0.88$

Answer:

1. $0.3090$
2. $0.5736$
3. $1.8040$
4. $0.5000$
5. $0.8660$
6. $0.5774$
7. $\frac{15}{17}$
8. $\frac{8}{15}$
9. $\frac{8}{17}$
10. $\frac{15}{17}$
11. $\frac{4}{5}$
12. $\frac{3}{4}$
13. $\frac{4}{5}$
14. $\frac{3}{5}$
15. $16.18$
16. $7.89$
17. $17.77$
18. $10.94$
19. $15.86$
20. $5.16$
21. $11.82$
22. $9.24$
23. $19.94$

Triangle 1: $x=5.75$, $y=7.62$
Triangle 2: $x=1.72$, $y=8.83$
Triangle 3: $x=1.80$, $y=0.88$