Question

19)
find the measure of the indicated angle to the nearest degree.
31
34
33
32

Explanation:

Step1: Identify trigonometric ratio

In the right triangle, we have the adjacent side (48) and the hypotenuse (57) relative to the indicated angle. So we use the cosine function: $\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{48}{57}$.

Step2: Calculate the angle

First, simplify $\frac{48}{57}\approx0.8421$. Then, take the inverse cosine: $\theta=\cos^{-1}(0.8421)$. Using a calculator, $\theta\approx32.6^\circ$, which rounds to 33 degrees. Wait, wait, let's recalculate. Wait, $\cos(32^\circ)\approx0.8480$, $\cos(33^\circ)\approx0.8387$. Wait, $\frac{48}{57}\approx0.8421$. Let's check: $\cos(32^\circ)\approx0.8480$, $\cos(33^\circ)\approx0.8387$. The value 0.8421 is closer to $\cos(32^\circ)$? Wait no, wait 0.8421 is between $\cos(32^\circ)$ (0.8480) and $\cos(33^\circ)$ (0.8387). Wait, 0.8421 - 0.8387 = 0.0034, 0.8480 - 0.8421 = 0.0059. So closer to 33? Wait no, wait cosine decreases as angle increases. So if $\cos(\theta)=0.8421$, let's calculate the angle. Using calculator: $\cos^{-1}(48/57)$. Let's compute 48 divided by 57: 48 ÷ 57 ≈ 0.842105. Then $\cos^{-1}(0.842105)\approx32.6$ degrees, which is approximately 33 degrees? Wait, no, wait maybe I mixed up adjacent and opposite. Wait, wait the triangle: the right angle, the side 48 is adjacent, hypotenuse 57. Wait, no, maybe it's the opposite side? Wait, no, the right angle, so the two legs: one is 48, hypotenuse 57. So the other leg is $\sqrt{57^2 - 48^2}=\sqrt{3249 - 2304}=\sqrt{945}\approx30.74$. Wait, maybe I used the wrong ratio. Wait, if the angle is at the end of the side 48, then the adjacent is 48, hypotenuse 57. So cosine. But maybe it's the opposite? Wait, no, the diagram: the right angle, one leg 48, hypotenuse 57, so the angle in question: let's see, the angle is at the vertex with the side 48 adjacent, so cosine. But let's recalculate $\cos^{-1}(48/57)$. 48 divided by 57 is approximately 0.8421. Using a calculator, $\cos^{-1}(0.8421)\approx32.6$ degrees, which is approximately 33? Wait, no, 32.6 is closer to 33? Wait, 32.6 is 0.6 degrees from 33, 0.4 degrees from 32? Wait, 32.6 - 32 = 0.6, 33 - 32.6 = 0.4. So closer to 33? Wait, no, 32.6 is 0.6 above 32, 0.4 below 33. So rounds to 33? But wait the options are 31,34,33,32. Wait, maybe I made a mistake. Wait, let's check with sine. If the opposite side is $\sqrt{57^2 - 48^2}=\sqrt{3249 - 2304}=\sqrt{945}\approx30.74$. Then sine of the angle would be $\frac{30.74}{57}\approx0.539$. Then $\sin^{-1}(0.539)\approx32.6$ degrees. Wait, no, that's the same as before. Wait, maybe the correct answer is 33? Wait, but let's check the options. Wait, maybe I messed up the ratio. Wait, maybe it's the adjacent and opposite. Wait, no, hypotenuse is 57, adjacent 48, so cosine. Let's use calculator: $\cos^{-1}(48/57)\approx32.6^\circ$, which is approximately 33 degrees. But wait the options include 33. Wait, but let's check again. Wait, 48 and 57: 48 is adjacent, hypotenuse 57. So $\cos(\theta)=48/57\approx0.8421$. Let's check the cosine values:

• $\cos(32^\circ)\approx0.8480$
• $\cos(33^\circ)\approx0.8387$

The difference between 0.8421 and 0.8480 is 0.0059, and between 0.8421 and 0.8387 is 0.0034. So 0.8421 is closer to 0.8387 (cos(33°))? Wait, no, cosine decreases as angle increases. So a larger angle has a smaller cosine. So 0.8421 is between cos(32°) (0.8480) and cos(33°) (0.8387). So the angle whose cosine is 0.8421 is between 32° and 33°. Since 0.8421 - 0.8387 = 0.0034 and 0.8480 - 0.8421 = 0.0059, it's closer to 33°? Wait, no, 0.0034 is less than 0.0059, so it's closer to 33°. So the angle is approximately 33…

Answer:

33 (corresponding to the option with 33)