How do I solve the equation:
3 = 2.1 sin[(pi/6)t - 5pi/6] + 3.2
To solve the equation 3 = 2.1 sin[(π/6)t - 5π/6] + 3.2, you can follow these steps:
Step 1: Move 3.2 to the other side of the equation
3 - 3.2 = 2.1 sin[(π/6)t - 5π/6]
Step 2: Simplify the left side of the equation
-0.2 = 2.1 sin[(π/6)t - 5π/6]
Step 3: Divide both sides of the equation by 2.1
(-0.2) / 2.1 = sin[(π/6)t - 5π/6]
Step 4: Simplify the left side of the equation
-0.09524 = sin[(π/6)t - 5π/6]
Step 5: Find the inverse sine (arcsin) of both sides to isolate the angle.
arcsin(-0.09524) = arcsin(sin[(π/6)t - 5π/6])
Step 6: Solve for the angle on the right side using the inverse sine function
-0.09613 = (π/6)t - 5π/6
Step 7: Add 5π/6 to both sides
-0.09613 + 5π/6 = (π/6)t
Step 8: Simplify the left side of the equation
4.3742 = (π/6)t
Step 9: Multiply both sides of the equation by 6/π to solve for t
(4.3742)(6/π) = t
Step 10: Calculate the value of t
t ≈ 13.91 (rounded to two decimal places)
So the solution to the equation 3 = 2.1 sin[(π/6)t - 5π/6] + 3.2 is t ≈ 13.91.
To solve the equation 3 = 2.1 sin[(π/6)t - 5π/6] + 3.2, you need to isolate the variable t. Here's how you can do it step by step:
Step 1: Move the constant term to the other side of the equation:
3 - 3.2 = 2.1 sin[(π/6)t - 5π/6]
Simplify the left side:
-0.2 = 2.1 sin[(π/6)t - 5π/6]
Step 2: Divide both sides of the equation by 2.1 to isolate the sine function:
(-0.2) / 2.1 = sin[(π/6)t - 5π/6]
Simplify the left side:
-0.095 = sin[(π/6)t - 5π/6]
Step 3: Take the inverse sine (also known as arcsine) of both sides to solve for the argument of the sine function:
arcsin(-0.095) = arcsin[sin[(π/6)t - 5π/6]]
Note: Inverse sine returns a principal value between -π/2 and π/2 radians (-90 and 90 degrees).
Step 4: Solve for the argument of the sine function:
(π/6)t - 5π/6 = arcsin(-0.095)
Step 5: Simplify the equation:
(π/6)t = arcsin(-0.095) + 5π/6
Step 6: Multiply both sides by 6/π to isolate t:
t = [arcsin(-0.095) + 5π/6] × 6/π
Now, use a calculator to evaluate [arcsin(-0.095) + 5π/6] × 6/π, and you will find the value of t that solves the equation.
3 = 2.1 sin[(π/6)t - 5π/6] + 3.2
2.1 sin[(π/6)t - 5π/6] = -0.2
sin[(π/6)t - 5π/6] = -0.095238
(π/6)t - 5π/6 = -.0953826
(π/6)t = 2.522611
t = 4.8178
and of course, there is another solution