Question

write an equation to find the missing side (x). show work. box the final answer. round to the nearest tenth.
1)

2)

3)

4)

5)

6)

7)

8)

9)

10)

Explanation:

Step1: Identify the trigonometric ratio (1)

For the first triangle, we have an angle of \(64^\circ\), adjacent side \(13\), and opposite side \(x\). So we use \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Thus, \(\tan(64^\circ)=\frac{x}{13}\).

Step2: Solve for \(x\) (1)

Multiply both sides by \(13\): \(x = 13\times\tan(64^\circ)\). Calculate \(\tan(64^\circ)\approx2.0503\), so \(x\approx13\times2.0503\approx26.7\).

Step3: Identify the trigonometric ratio (2)

For the second triangle, angle \(18^\circ\), adjacent side \(10\), opposite side \(x\). Use \(\tan(18^\circ)=\frac{x}{10}\).

Step4: Solve for \(x\) (2)

Multiply by \(10\): \(x = 10\times\tan(18^\circ)\). \(\tan(18^\circ)\approx0.3249\), so \(x\approx10\times0.3249\approx3.2\).

Step5: Identify the trigonometric ratio (3)

Third triangle: angle \(53^\circ\), hypotenuse \(20\), adjacent side \(x\). Use \(\cos(53^\circ)=\frac{x}{20}\) (since \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)).

Step6: Solve for \(x\) (3)

Multiply by \(20\): \(x = 20\times\cos(53^\circ)\). \(\cos(53^\circ)\approx0.6\), so \(x\approx20\times0.6 = 12.0\) (more accurately, \(\cos(53^\circ)\approx0.6018\), \(x\approx20\times0.6018\approx12.0\)).

Step7: Identify the trigonometric ratio (4)

Fourth triangle: angle \(54^\circ\), adjacent side \(11\), hypotenuse \(x\). Use \(\cos(54^\circ)=\frac{11}{x}\) (rearranged: \(x=\frac{11}{\cos(54^\circ)}\)).

Step8: Solve for \(x\) (4)

\(\cos(54^\circ)\approx0.5878\), so \(x\approx\frac{11}{0.5878}\approx18.7\).

Step9: Identify the trigonometric ratio (5)

Fifth triangle: angle \(26^\circ\), adjacent side \(16\), hypotenuse \(x\). Use \(\cos(26^\circ)=\frac{16}{x}\) (rearranged: \(x=\frac{16}{\cos(26^\circ)}\)).

Step10: Solve for \(x\) (5)

\(\cos(26^\circ)\approx0.8988\), so \(x\approx\frac{16}{0.8988}\approx17.8\).

Step11: Identify the trigonometric ratio (6)

Sixth triangle: angle \(60^\circ\), opposite side \(16\), adjacent side \(x\). Use \(\tan(60^\circ)=\frac{16}{x}\) (rearranged: \(x=\frac{16}{\tan(60^\circ)}\)).

Step12: Solve for \(x\) (6)

\(\tan(60^\circ)=\sqrt{3}\approx1.732\), so \(x\approx\frac{16}{1.732}\approx9.2\).

Step13: Identify the trigonometric ratio (7)

Seventh triangle: angle \(31^\circ\), hypotenuse \(20\), adjacent side \(x\). Use \(\cos(31^\circ)=\frac{x}{20}\).

Step14: Solve for \(x\) (7)

\(x = 20\times\cos(31^\circ)\). \(\cos(31^\circ)\approx0.8572\), so \(x\approx20\times0.8572\approx17.1\).

Step15: Identify the trigonometric ratio (8)

Eighth triangle: angle \(73^\circ\), opposite side \(14\), hypotenuse \(x\). Use \(\sin(73^\circ)=\frac{14}{x}\) (rearranged: \(x=\frac{14}{\sin(73^\circ)}\)).

Step16: Solve for \(x\) (8)

\(\sin(73^\circ)\approx0.9563\), so \(x\approx\frac{14}{0.9563}\approx14.6\).

Step17: Identify the trigonometric ratio (9)

Ninth triangle: angle \(20^\circ\), adjacent side \(15\), hypotenuse \(x\). Use \(\cos(20^\circ)=\frac{15}{x}\) (rearranged: \(x=\frac{15}{\cos(20^\circ)}\)).

Step18: Solve for \(x\) (9)

\(\cos(20^\circ)\approx0.9397\), so \(x\approx\frac{15}{0.9397}\approx15.9\).

Step19: Identify the trigonometric ratio (10)

Tenth triangle: angle \(23^\circ\), hypotenuse \(17\), opposite side \(x\). Use \(\sin(23^\circ)=\frac{x}{17}\).

Step20: Solve for \(x\) (10)

\(x = 17\times\sin(23^\circ)\). \(\sin(23^\circ)\approx0.3907\), so \(x\approx17\times0.3907\approx6.6\).

Answer:

1. \(\boxed{26.7}\)
2. \(\boxed{3.2}\)
3. \(\boxed{12.0}\)
4. \(\boxed{18.7}\)
5. \(\boxed{17.8}\)
6. \(\boxed{9.2}\)
7. \(\boxed{17.1}\)
8. \(\boxed{14.6}\)
9. \(\boxed{15.9}\)
10. \(\boxed{6.6}\)