Question

1. a right triangle with angle 28°, adjacent side 18, hypotenuse x. 8) a right triangle with hypotenuse 16, angle 21°, opposite side x. 9) a right triangle with angle 58°, opposite side 14, adjacent side x. 10) a right triangle with hypotenuse 13, angle 40°, adjacent side x.

Explanation:

Step 1: Problem 7 - Identify Trig Ratio

In the right triangle (7), we have the adjacent side to \(28^\circ\) as \(18\), and \(x\) is the hypotenuse. Use \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), so \(\cos(28^\circ)=\frac{18}{x}\).

Step 2: Solve for \(x\) (Problem 7)

Rearrange: \(x = \frac{18}{\cos(28^\circ)}\). Calculate \(\cos(28^\circ)\approx0.8829\), so \(x\approx\frac{18}{0.8829}\approx20.4\).

Step 3: Problem 8 - Identify Trig Ratio

In triangle (8), hypotenuse is \(16\), angle \(21^\circ\), \(x\) is opposite. Use \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), so \(\sin(21^\circ)=\frac{x}{16}\).

Step 4: Solve for \(x\) (Problem 8)

Rearrange: \(x = 16\sin(21^\circ)\). \(\sin(21^\circ)\approx0.3584\), so \(x\approx16\times0.3584\approx5.73\).

Step 5: Problem 9 - Identify Trig Ratio

In triangle (9), opposite side to \(58^\circ\) is \(14\), \(x\) is adjacent. Use \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), so \(\tan(58^\circ)=\frac{14}{x}\).

Step 6: Solve for \(x\) (Problem 9)

Rearrange: \(x=\frac{14}{\tan(58^\circ)}\). \(\tan(58^\circ)\approx1.6003\), so \(x\approx\frac{14}{1.6003}\approx8.75\).

Step 7: Problem 10 - Identify Trig Ratio

In triangle (10), hypotenuse \(13\), angle \(40^\circ\), \(x\) is adjacent. Use \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), so \(\cos(40^\circ)=\frac{x}{13}\).

Step 8: Solve for \(x\) (Problem 10)

Rearrange: \(x = 13\cos(40^\circ)\). \(\cos(40^\circ)\approx0.7660\), so \(x\approx13\times0.7660\approx9.96\).

Answer:

s:

1. \(x\approx20.4\)
2. \(x\approx5.73\)
3. \(x\approx8.75\)
4. \(x\approx9.96\)