Solve 8sin(pi/4x) = 6 for the four smallest positive solutions
Give your answers accurate to at least two decimal places, as a list separated by commas.
I got πx/4 = .84806
but how did you get πx/4 = 2.293, for the second value?
pi-.848
possibly?
pi-0.848062078981 = 2.293
Why did the math book look sad? Because it had too many problems!
Now, let's solve the equation 8sin(pi/4x) = 6.
Dividing both sides by 8, we get sin(pi/4x) = 6/8 = 3/4.
To find the four smallest positive solutions, we need to analyze the behavior of the sine function and determine where it equals 3/4.
First, let's find the reference angle. The angle whose sine is 3/4 is pi/3.
Since sin(x) = sin(pi - x), the solutions will be symmetrical around pi/4. Therefore, we're interested in finding an angle such that sin(x) = 3/4, where x lies between 0 and pi/2.
Using this information and the reference angle, we have two possible solutions:
1. x = pi/3
2. x = pi - (pi/3) = 2pi/3
Since we need four solutions, we can use the periodicity of the sine function to find additional solutions.
From x = pi/3, we can add 2pi to x repeatedly to obtain more solutions. However, we already have two solutions in the desired range, so we don't need to go beyond that.
Therefore, the four smallest positive solutions are:
x = pi/3, 2pi/3, pi + pi/3, and pi + 2pi/3.
Now, let's approximate these solutions accurate to at least two decimal places:
x ≈ 1.05, 2.09, 4.19, and 5.24.
Remember, laughter is the best solution to any problem!
To solve the equation 8sin(pi/4x) = 6 for the four smallest positive solutions, we can follow these steps:
1. Divide both sides of the equation by 8 to isolate the sine function:
sin(pi/4x) = 6/8
sin(pi/4x) = 3/4
2. Take the inverse sine (sin^(-1)) of both sides to eliminate the sin function and find the angle:
pi/4x = sin^(-1)(3/4)
3. To find the four smallest positive solutions, we need to find the values of x that satisfy the equation. Since sine has a period of 2π, we need to find the range of angles for which sin(pi/4x) is equal to 3/4.
We can start by finding the principal angle, which is the angle between -π/2 and π/2 that has a sine value of 3/4:
sin(pi/4) = 3/4
pi/4 ≈ 0.7854
4. Since sin is positive in both the first and second quadrants, we can extend the principal angle by adding multiples of 2π to find other angles with the same sine value.
The four smallest positive solutions can be found by using the following formula:
x = [(2πn + π/4)/(π/4)] where n = 0, 1, 2, 3
5. Substitute the values of n into the formula to find the four smallest positive solutions:
For n = 0:
x = [(2π(0) + π/4)/(π/4)] ≈ 1.0000
For n = 1:
x = [(2π(1) + π/4)/(π/4)] ≈ 4.0000
For n = 2:
x = [(2π(2) + π/4)/(π/4)] ≈ 7.0000
For n = 3:
x = [(2π(3) + π/4)/(π/4)] ≈ 10.0000
Therefore, the four smallest positive solutions to the equation 8sin(pi/4x) = 6 are approximately 1.00, 4.00, 7.00, and 10.00.
8sin(pi/4x) = 6
sin ((π/4)x) = .75
πx/4 = .84806 or πx/4 = 2.2935
x = 1.07979 or 2.9202
the period of the original sine curve is 2π/(π/4) = 8
so by adding 8 to the above we get the other two x values
x = 1.07979 , 2.2935 , 9.079 , 10.9202