Question

58 what is the value of $\theta$? (show all work on paper): right triangle with horizontal leg 14, hypotenuse 20, angle $\theta$ at the bottom left vertex a $25^\circ$ b $44.4^\circ$ c $62.8^\circ$ d $30^\circ$

Explanation:

Step1: Identify trigonometric ratio

In the right triangle, the adjacent side to \( \theta \) is 14? Wait, no, wait. Wait, the right angle is at the top left. So the side adjacent to \( \theta \) is the vertical leg? Wait, no. Wait, the hypotenuse is 20, and the horizontal leg (adjacent to \( \theta \))? Wait, no, let's label the triangle. The right angle is at the top left, so the vertices are: top left (right angle), top right (length 14), bottom left (angle \( \theta \)), and hypotenuse is 20 (connecting top right to bottom left). So the side adjacent to \( \theta \) is the vertical leg? Wait, no. Wait, the horizontal leg (top side) is 14, which is opposite to \( \theta \)? Wait, no. Wait, in a right triangle, for angle \( \theta \) at the bottom left: the opposite side is the top horizontal side (length 14), the adjacent side is the vertical leg (let's call it \( a \)), and hypotenuse is 20. So we can use sine: \( \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{14}{20} \).

Step2: Calculate \( \theta \)

So \( \sin\theta = \frac{14}{20} = 0.7 \). Then \( \theta = \arcsin(0.7) \). Let's calculate that. Using a calculator, \( \arcsin(0.7) \approx 44.4^\circ \)? Wait, no, wait: wait, 14 is opposite? Wait, maybe I mixed up opposite and adjacent. Wait, let's re-examine the triangle. The right angle is at the top left. So the sides: top side (horizontal) is 14, left side (vertical) is adjacent to \( \theta \), hypotenuse is 20 (connecting top right to bottom left). So angle \( \theta \) is at the bottom left. So the side opposite to \( \theta \) is the top horizontal side (14), hypotenuse is 20. So \( \sin\theta = \frac{14}{20} = 0.7 \). Then \( \theta = \arcsin(0.7) \approx 44.4^\circ \)? Wait, but let's check with cosine. Wait, maybe I had it wrong. Wait, if the horizontal side is 14, and hypotenuse is 20, then \( \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \). Wait, no, adjacent to \( \theta \) is the vertical leg. Wait, maybe I made a mistake. Wait, let's use cosine. Wait, the adjacent side to \( \theta \) is the vertical leg, and the hypotenuse is 20. Wait, no, the horizontal leg (14) is opposite to \( \theta \), so \( \sin\theta = \frac{14}{20} \). Let's compute \( \arcsin(14/20) \). 14 divided by 20 is 0.7. \( \arcsin(0.7) \approx 44.4^\circ \)? Wait, but 0.7 sine: let's check with calculator. \( \sin(44.4^\circ) \approx \sin(44^\circ) \approx 0.6947 \), which is close to 0.7. So that's correct. So \( \theta \approx 44.4^\circ \), which is option B.

Answer:

B. \( 44.4^\circ \)