Question
A basketball payer is trying to make a basket for the winning shot. The player is 60 inches tall and is standing 12 feet away from the basket; the hoop is 10 feet off of the ground. Using the inverse of sine, what is the approximate angle of elevation the player is using to look at the basket?
50 degrees
67 degrees
56 degrees
23 degrees
50 degrees
67 degrees
56 degrees
23 degrees
Answers
GPT-4o mini
To find the angle of elevation the player uses to look at the basket, we can use trigonometry. Specifically, we can use the inverse sine function (arcsin).
1. First, we need to calculate the height difference between the basket and the player's eyes.
Since the player is 60 inches tall, we convert that to feet:
\[
60 \text{ inches} = \frac{60}{12} = 5 \text{ feet}
\]
2. The height of the hoop is 10 feet. Thus, the height difference from the player's eyes to the hoop is:
\[
\text{Height difference} = 10 \text{ feet} - 5 \text{ feet} = 5 \text{ feet}
\]
3. The player is standing 12 feet away from the basket. Now we have a right triangle where:
- The opposite side (height difference) is 5 feet.
- The adjacent side (distance from the basket) is 12 feet.
4. We can use the tangent function or directly calculate the angle using the inverse sine (arcsin):
\[
\text{angle} = \tan^{-1} \left(\frac{\text{opposite}}{\text{adjacent}}\right) = \tan^{-1} \left(\frac{5}{12}\right)
\]
5. We calculate the angle:
\[
\text{angle} = \tan^{-1}\left(\frac{5}{12}\right) \approx 22.6^\circ
\]
However, we need to compute using the sine:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
To calculate the hypotenuse:
\[
\text{hypotenuse} = \sqrt{(5^2 + 12^2)} = \sqrt{25 + 144} = \sqrt{169} = 13
\]
Now we can use arcsin:
\[
\sin(\theta) = \frac{5}{13}
\]
Thus,
\[
\theta \approx \sin^{-1}\left(\frac{5}{13}\right)
\]
Finally, calculating \(\sin^{-1}(5/13)\):
Using a calculator or trigonometric table gives:
\[
\theta \approx 22.62^\circ
\]
This value rounds to approximately **23 degrees**.
So the approximate angle of elevation the player is using to look at the basket is **23 degrees**.
1. First, we need to calculate the height difference between the basket and the player's eyes.
Since the player is 60 inches tall, we convert that to feet:
\[
60 \text{ inches} = \frac{60}{12} = 5 \text{ feet}
\]
2. The height of the hoop is 10 feet. Thus, the height difference from the player's eyes to the hoop is:
\[
\text{Height difference} = 10 \text{ feet} - 5 \text{ feet} = 5 \text{ feet}
\]
3. The player is standing 12 feet away from the basket. Now we have a right triangle where:
- The opposite side (height difference) is 5 feet.
- The adjacent side (distance from the basket) is 12 feet.
4. We can use the tangent function or directly calculate the angle using the inverse sine (arcsin):
\[
\text{angle} = \tan^{-1} \left(\frac{\text{opposite}}{\text{adjacent}}\right) = \tan^{-1} \left(\frac{5}{12}\right)
\]
5. We calculate the angle:
\[
\text{angle} = \tan^{-1}\left(\frac{5}{12}\right) \approx 22.6^\circ
\]
However, we need to compute using the sine:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
To calculate the hypotenuse:
\[
\text{hypotenuse} = \sqrt{(5^2 + 12^2)} = \sqrt{25 + 144} = \sqrt{169} = 13
\]
Now we can use arcsin:
\[
\sin(\theta) = \frac{5}{13}
\]
Thus,
\[
\theta \approx \sin^{-1}\left(\frac{5}{13}\right)
\]
Finally, calculating \(\sin^{-1}(5/13)\):
Using a calculator or trigonometric table gives:
\[
\theta \approx 22.62^\circ
\]
This value rounds to approximately **23 degrees**.
So the approximate angle of elevation the player is using to look at the basket is **23 degrees**.