The image contains two trigonometry problems involving right triangles.

1. Problem 1:

   • A post of height 20 meters is to be supported by a cable that forms a 30° angle with the ground.
   • The task is to find the length of the cable.

2. Problem 2:

   • We have a tree or a post where, from a distance of 8 meters from the base, the top is visible at an angle of elevation of 36.87°.
   • The task is to calculate the height, $h$, of the tree.

Solving:

Problem 1:

We are given a right triangle where:

• The height of the post (opposite side) is 20 meters.
• The angle between the ground and the cable is $30^\circ$.

We use the sine function: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{20}{\text{cable length}}$ Substitute $\theta = 30^\circ$: $\sin(30^\circ) = \frac{20}{\text{cable length}}$ Since $\sin(30^\circ) = 0.5$, we have: $0.5 = \frac{20}{\text{cable length}}$ $\text{Cable length} = \frac{20}{0.5} = 40 \text{ meters}$ So, the cable length is 40 meters.

Problem 2:

We are given:

• The distance from the base of the tree/post is 8 meters (adjacent side).
• The angle of elevation is $36.87^\circ$.

We use the tangent function: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{8}$ Substitute $\theta = 36.87^\circ$: $\tan(36.87^\circ) = \frac{h}{8}$ Since $\tan(36.87^\circ) \approx 0.75$, we have: $0.75 = \frac{h}{8}$ $h = 0.75 \times 8 = 6 \text{ meters}$ So, the height of the tree/post is 6 meters.

Would you like further details on any part of the solution?

Related Questions:

1. How would the solution change if the angle in problem 1 were 45°?
2. Can we use the cosine function to find the horizontal distance in problem 1?
3. How do you calculate the length of the shadow cast by the tree in problem 2?
4. What would happen if the angle of elevation in problem 2 were increased?
5. How can you verify the results of both problems using the Pythagorean theorem?

Tip: In right triangles, knowing one angle and one side allows you to solve for all other sides using trigonometric ratios like sine, cosine, and tangent.