Solve sin(x) = 0.64 on 0 less than or equal to x less than 2pi
There are two solutions, A and B, with A < B
Give your answers accurate to 3 decimal places
set your calculator to radians and find
sin^-1 (.64)
you should get appr .6945
proceed like I just showed you in the 2 previous posts.
To solve the equation sin(x) = 0.64, follow these steps:
Step 1: Identify the range
The given range is 0 ≤ x < 2π (or 0 ≤ x < 6.283).
Step 2: Determine the reference angle
Since the sine function is positive in the first and second quadrants, we can find the reference angle for 0.64 by using the inverse sine function. We get:
sin^(-1)(0.64) ≈ 0.692 radians
Step 3: Find the first solution (A)
The first solution occurs in the first quadrant, where the sine function is positive. Therefore,
A = 0.692 radians ≈ 0.692
Step 4: Find the second solution (B)
The second solution occurs in the second quadrant. To find the angle in the second quadrant with the same reference angle, subtract the reference angle from π radians (180 degrees):
B = π − 0.692 radians ≈ 2.449
So, the solutions to the equation sin(x) = 0.64 in the given range are approximately:
A ≈ 0.692
B ≈ 2.449
To solve the equation sin(x) = 0.64 on the interval 0 ≤ x ≤ 2π, we need to find the values of x that satisfy this equation. Here's how you can proceed:
Step 1: Start by finding the principal (or inverse) sine of 0.64. The principal sine is denoted as sin^(-1)(0.64) or arcsin(0.64). This can be done using a scientific calculator or by using an online calculator that supports trigonometric functions. By evaluating this, you will find the principal angle (θ) that gives sin(θ) = 0.64.
arcsin(0.64) ≈ 0.687
Step 2: Since we are looking for solutions on the interval 0 ≤ x ≤ 2π, we can proceed to find the solutions. There are two main solutions within this interval, but there can be infinitely many solutions when considering the periodic nature of sine.
One solution occurs when x = arcsin(0.64), which is approximately 0.687.
Step 3: To find the second solution, we need to take the domain of x into account. Since the sine function is periodic with a period of 2π, we can add 2π to the first solution to find the second solution.
x = 0.687 + 2π ≈ 6.929
Step 4: Verify that these solutions fall within the given range, which is 0 ≤ x ≤ 2π. In this case, both solutions satisfy this condition.
Therefore, the two solutions, A and B, with A < B, are approximately:
Solution A: x ≈ 0.687
Solution B: x ≈ 6.929
These solutions accurately represent the answers to the equation sin(x) = 0.64 on the interval 0 ≤ x ≤ 2π, rounded to 3 decimal places.