Solve sin(x)=−0.89 on 0≤x<2π
There are two solutions, A and B, with A < B
To solve the equation sin(x) = -0.89 on the interval 0 ≤ x < 2π, we can use the inverse sine function (also known as arcsine).
Step 1: Take the inverse sine of both sides of the equation:
arcsin(sin(x)) = arcsin(-0.89)
Step 2: Simplify the left side using the fact that the inverse sine undoes the sine function:
x = arcsin(-0.89)
Step 3: Use a calculator or a table of trigonometric values to find the principal value of arcsin(-0.89). The principal value is the angle whose sine is equal to -0.89. You should get a value close to -61.37 degrees or -1.07 radians.
Step 4: We are given the interval 0 ≤ x < 2π, which means we are looking for solutions within one full revolution (0 to 2π). Since the principal value we found (-1.07 radians) is negative, we need to find the second value within the interval that is equivalent to it.
Step 5: Add 2π (or 360 degrees) to the principal value to find the second solution:
x = -1.07 + 2π
Step 6: Simplify the second solution:
x = 5.22 radians (approximately)
So, the two solutions, A and B, with A < B, are approximately x = -1.07 radians and x = 5.22 radians.