To find the angle formed between the pole and the string, we can use the cosine function from trigonometry.
In this scenario, we can consider the situation as a right triangle where:
- The height of the pole (8 feet) represents the opposite side.
- The length of the string (12 feet) represents the hypotenuse.
Using the cosine function:
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \] In our case, we are looking for θ, and we will use the Pythagorean theorem to first find the adjacent side.
- Calculate the adjacent side using the Pythagorean theorem:
\[ \text{adjacent}^2 + \text{opposite}^2 = \text{hypotenuse}^2 \] \[ \text{adjacent}^2 + 8^2 = 12^2 \] \[ \text{adjacent}^2 + 64 = 144 \] \[ \text{adjacent}^2 = 144 - 64 = 80 \] \[ \text{adjacent} = \sqrt{80} \approx 8.94 \text{ feet} \]
- Now we can calculate the angle θ:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{8}{12} = \frac{2}{3} \]
Using the inverse sine (arcsin) to find the angle:
\[ \theta = \arcsin\left(\frac{2}{3}\right) \]
Using a calculator to find the angle gives us approximately:
\[ \theta \approx 42° \]
Therefore, the approximate angle formed between the pole and the string is:
42°