Question

find the measure of the indicated angle to the nearest degree.
44)
a) 32° b) 40°
c) 50° d) 81°
45)
a) 22° b) 68°
c) 82° d) 47°
46)
a) 70° b) 22°
c) 20° d) 68°
find the area of each.
47)
48)
49)
50)

Explanation:

Problem 44:

Step1: Identify triangle type and trigonometric ratio

This is a right triangle. We know the adjacent side (44) and the hypotenuse (52) relative to the unknown angle. So we use the cosine function: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{44}{52}$.

Step2: Calculate the angle

First, simplify $\frac{44}{52} \approx 0.8462$. Then, take the arccosine: $\theta = \arccos(0.8462) \approx 32^\circ$? Wait, no, wait. Wait, adjacent and opposite? Wait, the legs are 52 and 44? Wait, no, the right angle is between the two legs. Wait, the side labeled 52 and 44 are the legs? Wait, no, the hypotenuse would be the side opposite the right angle. Wait, maybe I mixed up. Wait, the triangle has a right angle, and the two legs are 52 and 44? Wait, no, the side with the question mark is an acute angle. Let's re-examine: the sides are 52 (one leg), 44 (another leg), and the hypotenuse? Wait, no, in a right triangle, the hypotenuse is the longest side. Wait, 52 and 44 are legs? Then the hypotenuse would be $\sqrt{52^2 + 44^2} \approx \sqrt{2704 + 1936} = \sqrt{4640} \approx 68.12$, but that's not one of the sides. Wait, maybe the side labeled 52 is the hypotenuse? Wait, the diagram: the right angle is between the side of length 44 and the hypotenuse? No, the right angle is at the vertex where the two legs meet. So the legs are 44 and the other leg, and the hypotenuse is 52? Wait, no, 52 is longer than 44, but if it's a leg, the hypotenuse would be longer. Wait, maybe I got adjacent and opposite wrong. Let's use sine: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$. If the opposite side to the angle is 52? No, that can't be, because hypotenuse is the longest side. Wait, maybe the sides are: the leg adjacent to the angle is 44, and the leg opposite is 52? Wait, then $\tan(\theta) = \frac{52}{44} \approx 1.1818$, so $\theta = \arctan(1.1818) \approx 49.7^\circ \approx 50^\circ$. Ah, that makes sense. So I had adjacent and opposite mixed. So the angle's opposite side is 52, adjacent is 44. So $\tan(\theta) = \frac{52}{44} \approx 1.1818$, so $\theta \approx 50^\circ$, which is option C.

Step1: Identify triangle type and trigonometric ratio

Right triangle. We know the adjacent side (39) and the hypotenuse (42) relative to the unknown angle. So $\cos(\theta) = \frac{39}{42} \approx 0.9286$.

Step2: Calculate the angle

Take the arccosine: $\theta = \arccos(0.9286) \approx 21.7^\circ \approx 22^\circ$? Wait, no, wait. Wait, the leg is 39, hypotenuse 42. Wait, or is the 39 the opposite side? Wait, the right angle is at the top, so the vertical leg is 39, hypotenuse 42. So the angle at the top right: adjacent side is 39, hypotenuse 42? No, adjacent would be the leg adjacent to the angle, which is the vertical leg (39), and the hypotenuse is 42. So $\cos(\theta) = \frac{39}{42} \approx 0.9286$, so $\theta \approx 22^\circ$, which is option A? Wait, but let's check with sine: opposite side would be the horizontal leg? Wait, no, the triangle has a right angle, so the two legs are 39 (vertical) and the horizontal leg (let's call it x), and hypotenuse 42. So $x = \sqrt{42^2 - 39^2} = \sqrt{1764 - 1521} = \sqrt{243} \approx 15.59$. Then, the angle at the top right: opposite side is x (15.59), adjacent is 39. So $\tan(\theta) = \frac{15.59}{39} \approx 0.3997$, so $\theta \approx 21.8^\circ \approx 22^\circ$, which is option A.

Step1: Identify triangle type and trigonometric ratio

Right triangle. We know the opposite side (22) and the hypotenuse (60) relative to the unknown angle. So use sine: $\sin(\theta) = \frac{22}{60} \approx 0.3667$.

Step2: Calculate the angle

Take the arcsine: $\theta = \arcsin(0.3667) \approx 21.5^\circ \approx 20^\circ$ (option C) or 70°? Wait, wait, the angle at the top: the side opposite is 22, hypotenuse 60. Wait, or is the angle at the bottom? Wait, the right angle is at the bottom right, so the angle at the top: opposite side is 22, hypotenuse 60. So $\sin(\theta) = 22/60 ≈ 0.3667$, so $\theta ≈ 21.5° ≈ 20°$ (option C) or the other angle: 90 - 20 = 70°, which is option A. Wait, let's check: the two acute angles add up to 90°. If one angle is ≈20°, the other is ≈70°. Let's use cosine for the angle at the bottom: adjacent side is 22, hypotenuse 60? No, adjacent to the bottom angle is 22, opposite is the horizontal leg. Wait, no, the side labeled 22 is vertical, 60 is hypotenuse. So the angle at the top: $\sin(\theta) = 22/60 ≈ 0.3667$, so $\theta ≈ 21.5° ≈ 20°$ (option C), but the angle at the bottom would be 90 - 20 = 70°, which is option A. Wait, the diagram: the triangle has a right angle, side 22 (vertical leg), hypotenuse 60, and the angle at the top is the one with the question mark? Wait, the label "22" is next to the vertical leg, and "60" is the hypotenuse. So the angle at the top: opposite side is the horizontal leg, adjacent is 22? No, I'm confused. Wait, let's use the other angle. Let's calculate the angle at the bottom: adjacent side is 22, hypotenuse 60? No, adjacent to the bottom angle is the horizontal leg, opposite is 22. Wait, no, let's use Pythagoras to find the horizontal leg: $x = \sqrt{60^2 - 22^2} = \sqrt{3600 - 484} = \sqrt{3116} ≈ 55.82$. Then, the angle at the bottom: $\cos(\theta) = 22/60 ≈ 0.3667$? No, adjacent is 22, hypotenuse 60? No, adjacent to the bottom angle is 22, hypotenuse 60? No, adjacent is the horizontal leg? Wait, I think I made a mistake. Let's start over. The right triangle has legs: vertical leg = 22, hypotenuse = 60. So the angle at the top (let's call it θ) has: opposite side = vertical leg = 22? No, opposite side would be the horizontal leg. Wait, no, the right angle is between the vertical leg (22) and the horizontal leg (let's say x), so hypotenuse is 60. So x = √(60² - 22²) ≈ 55.82. Then, the angle at the top (θ) has: adjacent side = 22, opposite side = x ≈ 55.82. So $\tan(\theta) = x/22 ≈ 55.82/22 ≈ 2.537$, so $\theta ≈ 68.5° ≈ 70°$ (option A). Ah, there we go! I had the opposite and adjacent mixed. So the angle at the top: adjacent is 22, opposite is x ≈ 55.82, so $\tan(\theta) = 55.82/22 ≈ 2.537$, so $\theta ≈ 68.5° ≈ 70°$, which is option A.

Answer:

C) $50^\circ$

Problem 45: