Question

unit: trigonometry progress: the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer. what is the value of y to the nearest hundredth? right triangle with hypotenuse 10, angle 48°, side y opposite? wait, no, right angle, so angle 48°, hypotenuse 10, side y adjacent? wait, the triangle has a right angle, angle 48°, hypotenuse 10, and side y. options: 11.11, 13.46, 7.43, 6.69

Explanation:

Step1: Identify trigonometric ratio

In the right triangle, we have the hypotenuse (10) and we need to find the opposite side to the 48° angle (y). The sine function relates the opposite side and the hypotenuse: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$.

Step2: Set up the equation

For $\theta = 48^\circ$, $\sin(48^\circ) = \frac{y}{10}$.

Step3: Solve for y

Multiply both sides by 10: $y = 10 \times \sin(48^\circ)$.

Step4: Calculate the value

Using a calculator, $\sin(48^\circ) \approx 0.7431$. Then $y \approx 10 \times 0.7431 = 7.431$, which rounds to 7.43. Wait, no, wait—wait, the angle is 48°, and the side y: wait, maybe I mixed up opposite and adjacent? Wait, no, the right angle, the angle given is 48°, so the side y: let's check the triangle again. The right angle, the angle at the top is 48°, so the side opposite to 48° is y? Wait, no, the hypotenuse is 10, the angle is 48°, so the side adjacent to 48°? Wait, no, the right angle is at the left, so the triangle has vertices: right angle at left, angle 48° at the top, so the sides: hypotenuse is the side opposite the right angle, so the hypotenuse is 10. Then the angle at the top is 48°, so the side opposite to 48° is the vertical side (y)? Wait, no, the left side is y, the bottom side is adjacent, the left side is opposite to the 48° angle? Wait, no, the angle at the top is 48°, so the side opposite to 48° is the left side (y), and the hypotenuse is 10. So $\sin(48°) = y/10$, so $y = 10 \sin(48°)$. Wait, but when I calculate $\sin(48°)$: let's use calculator. $\sin(48) \approx 0.7431$, so 100.7431 = 7.431, which is 7.43, but the option is 7.43? Wait, the options are 11.11, 13.46, 7.43, 6.69. Wait, maybe I made a mistake. Wait, maybe it's cosine? Wait, no, if the angle is 48°, and the side y is adjacent? Wait, no, the right angle is at the left, so the sides: horizontal side (adjacent to 48°), vertical side (opposite to 48°), hypotenuse 10. Wait, maybe the angle is at the bottom? Wait, the diagram: right angle at the top left, angle 48° at the top right, hypotenuse 10, and y is the vertical side (left side). So angle at top right is 48°, so the side opposite to 48° is the left side (y), hypotenuse 10. So $\sin(48°) = y/10$, so $y = 10 \sin(48°) \approx 7.43$. But the option is 7.43? Wait, the options are 11.11, 13.46, 7.43, 6.69. So 7.43 is one of the options (7.43 is 7.43, which is 7.43, matching the third option). Wait, maybe I miscalculated earlier. Let me check again. $\sin(48°) \approx 0.7431$, so 100.7431 = 7.431, which is 7.43 when rounded to the nearest hundredth. So the correct answer is 7.43, which is the third option.

Answer:

7.43 (corresponding to the option 7.43, which is the third option in the list: 7.43)