Question

a building casts a shadow reaching 13 feet from the base of the building, with a diagonal distance of 15 feet from the top of the building. using the inverse of sine, what is the approximate angle formed between the top of the building and the shadow? (1 point) 30° 60° 90° 45°

Explanation:

Step1: Identify the triangle sides

We have a right triangle where the adjacent side to the angle (let's call it $\theta$) is the shadow length, $a = 13$ feet, and the hypotenuse is the diagonal distance, $c = 15$ feet. We use the cosine function? Wait, no, the problem says to use the inverse of sine. Wait, sine of an angle in a right triangle is opposite over hypotenuse. Wait, maybe I misread. Wait, the angle between the top of the building and the shadow: so the shadow is the adjacent side? Wait, no. Let's visualize: the building is vertical (opposite side), the shadow is horizontal (adjacent side), and the diagonal is the hypotenuse. The angle between the top of the building (so from the top to the end of the shadow) and the shadow: so the angle at the end of the shadow, between the shadow (adjacent) and the hypotenuse (line from top of building to end of shadow). Wait, no, maybe the angle at the top? Wait, no, the problem says "the angle formed between the top of the building and the shadow". Wait, maybe the angle at the end of the shadow, between the shadow (adjacent) and the line from top to shadow (hypotenuse). Wait, but the problem says to use the inverse of sine. So sine of the angle is opposite over hypotenuse. The opposite side to the angle (let's say $\theta$) would be the height of the building. Wait, but we can find the height first using Pythagoras: $h = \sqrt{15^2 - 13^2} = \sqrt{225 - 169} = \sqrt{56} \approx 7.483$ feet. Then, sine of $\theta$ (the angle at the end of the shadow, between the shadow and the hypotenuse) would be opposite over hypotenuse, which is $h / 15 \approx 7.483 / 15 \approx 0.4989$. Then, $\theta = \arcsin(0.4989) \approx 30^\circ$ (since $\sin(30^\circ) = 0.5$). Wait, that's close. Let's check: $\sin(30^\circ) = 0.5$, and our value is approximately 0.4989, which is very close to 0.5, so the angle is approximately $30^\circ$. Wait, but let's confirm. Alternatively, maybe the angle is at the top? No, the shadow is at the base. Wait, the problem says "the angle formed between the top of the building and the shadow". So the two sides forming the angle are: one is the shadow (from base to end, horizontal), and the other is from top of building to end of shadow (hypotenuse). Wait, no, the top of the building to the shadow: the shadow is on the ground, so the angle between the top of the building (vertical) and the shadow? No, that doesn't make sense. Wait, maybe the angle of elevation from the end of the shadow to the top of the building. Wait, the angle of elevation $\theta$ has opposite side (height of building) and adjacent side (shadow length). But the problem says to use inverse of sine. So sine of $\theta$ is opposite over hypotenuse. So opposite is height, hypotenuse is 15. So $\sin(\theta) = h / 15$. We found $h \approx 7.483$, so $\sin(\theta) \approx 7.483 / 15 \approx 0.4989$, so $\theta = \arcsin(0.4989) \approx 30^\circ$, since $\sin(30^\circ) = 0.5$. So the angle is approximately $30^\circ$.

Step2: Calculate the angle using arcsin

We have $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{15^2 - 13^2}}{15} = \frac{\sqrt{56}}{15} \approx \frac{7.483}{15} \approx 0.4989$. Then, $\theta = \arcsin(0.4989) \approx 30^\circ$ (since $\sin(30^\circ) = 0.5$ and 0.4989 is very close to 0.5).

Answer:

30°