Question

Find all angles between 0° and 180° that satisfy the given equation. Round your answer(s) to one decimal place. (Enter your answers as a comma-separated list. Enter your answer in degrees.)

sin() = 5/7

Answer 1

The angles between 0° and 180° that satisfy sin(θ) = 5/7 are approximately 45.6° and 134.4°. The first angle is found using the inverse sine function, and the second angle is the complement to 180° of the first angle.

Step-by-step explanation:

To find the angles between 0° and 180° that satisfy the equation sin(θ) = 5/7, we first recall that sin is positive in the first and second quadrants (0° to 180°). We then find the principal angle by using the inverse sine function, often denoted as sin-1 or arcsin. The principal angle θ1 is obtained by calculating sin-1(5/7).

θ1 = sin-1(5/7) ≈ 45.6°

To find the second solution, θ2, which lies in the second quadrant, we use the fact that the sine function is symmetric with respect to 180°. We calculate θ2 as 180° - θ1.

θ2 = 180° - 45.6° ≈ 134.4°

The two angles that satisfy the equation sin(θ) = 5/7 between 0° and 180°, rounded to one decimal place, are 45.6° and 134.4°.