solve 3sin(pi/4*x)=2 for the four smallest positive solutions
Why did the sine cross the road? Because it was trying to solve an equation! Okay, let's get serious now.
To solve the equation 3sin(pi/4*x) = 2, we can solve for sin(pi/4*x) first and then find the values of x.
Dividing both sides by 3, we get sin(pi/4*x) = 2/3.
Now, let's find the reference angle, which is the angle in the first quadrant whose sine is 2/3. We can use the inverse sine function (also known as arcsin) to do that: arcsin(2/3) ≈ 0.9273 radians.
Since we're looking for the smallest positive solutions, we need to consider all x values that give us sin(pi/4*x) = 2/3 and fall within the range of 0 to 2π (one full revolution).
The four smallest positive solutions are:
1) x = 4(0.9273)/(π) ≈ 1.183 radians.
2) x = [(4(0.9273)+(2π) ]/π ≈ 2.959 radians.
3) x = [(4(0.9273)+(4π) ]/π ≈ 4.735 radians.
4) x = [(4(0.9273)+(6π) ]/π ≈ 7.511 radians.
So the four smallest positive solutions to the equation 3sin(pi/4*x) = 2 are approximately:
1) x ≈ 1.183 radians.
2) x ≈ 2.959 radians.
3) x ≈ 4.735 radians.
4) x ≈ 7.511 radians.
Now I hope this solution has brought a smile to your face, even if it didn't make you laugh out loud! 🤡
To solve the equation 3sin(pi/4*x) = 2, we can follow these steps:
Step 1: Divide both sides of the equation by 3:
sin(pi/4*x) = 2/3
Step 2: Find the inverse sine of both sides to eliminate the sine function:
pi/4*x = arcsin(2/3)
Step 3: Solve for x by multiplying both sides by 4/pi:
x = (4/pi) * arcsin(2/3)
Step 4: Calculate the numerical value of arcsin(2/3):
Using a calculator or table, we find that arcsin(2/3) is approximately 0.7297 radians or 41.81 degrees.
Step 5: Substitute the numerical value back into the equation for x:
x = (4/pi) * 0.7297
Step 6: Calculate the value of x using the approximate value of pi (3.14159):
x ≈ (4/3.14159) * 0.7297
x ≈ 0.9247 radians or approximately 52.99 degrees.
So, the first positive solution is x ≈ 0.9247 or approximately 52.99 degrees.
Step 7: To find the second positive solution, add the period of the sine function, which is 2pi (or 360 degrees), to the previous solution:
x ≈ 0.9247 + 2pi radians or approximately 52.99 + 360 degrees
x ≈ 0.9247 + 6.283 radians or approximately 52.99 + 360 degrees
x ≈ 7.2077 radians or approximately 412.99 degrees.
Step 8: Repeat steps 5, 6, and 7 to find the next two positive solutions:
Second positive solution:
x ≈ (4/3.14159) * 0.7297 + 2pi
x ≈ 0.9247 + 6.283
x ≈ 7.2077 radians or approximately 412.99 degrees.
Third positive solution:
x ≈ (4/3.14159) * 0.7297 + 4pi
x ≈ 0.9247 + 12.566
x ≈ 13.3777 radians or approximately 767.99 degrees.
Fourth positive solution:
x ≈ (4/3.14159) * 0.7297 + 6pi
x ≈ 0.9247 + 18.850
x ≈ 19.5477 radians or approximately 1122.99 degrees.
Therefore, the four smallest positive solutions to the equation 3sin(pi/4*x) = 2 are approximately:
x ≈ 0.9247 radians or approximately 52.99 degrees,
x ≈ 7.2077 radians or approximately 412.99 degrees,
x ≈ 13.3777 radians or approximately 767.99 degrees, and
x ≈ 19.5477 radians or approximately 1122.99 degrees.
To solve the equation 3sin(pi/4*x) = 2 for the four smallest positive solutions, we need to isolate the variable x.
Step 1: Divide both sides of the equation by 3 to isolate the sine function:
sin(pi/4*x) = 2/3
Step 2: Take the inverse sine (arcsine) of both sides to cancel out the sine function:
pi/4*x = arcsin(2/3)
Step 3: Multiply both sides of the equation by 4/pi to solve for x:
x = (4/pi) * arcsin(2/3)
To find the four smallest positive solutions, we can use a calculator or mathematical software that can evaluate the arcsine function. Let's proceed with the calculation assuming you have access to a calculator:
Step 4: Calculate the value of arcsin(2/3) using a calculator:
arcsin(2/3) ≈ 0.7297 (in radians)
Step 5: Substitute the value of arcsin(2/3) into the equation for x:
x = (4/pi) * 0.7297 ≈ 0.9250
Therefore, the first smallest positive solution is x ≈ 0.9250.
Step 6: To find the other three smallest positive solutions, we add integer multiples of the period of the sine function, which is 2π. So the remaining solutions can be found with the following formula:
x = 0.9250 + n*2π, where n is an integer.
Substituting n = 1, 2, and 3 into the equation above will give us the other three solutions. Let's calculate them:
For n = 1:
x = 0.9250 + 1*2π ≈ 7.1989
For n = 2:
x = 0.9250 + 2*2π ≈ 13.4729
For n = 3:
x = 0.9250 + 3*2π ≈ 19.7468
Therefore, the four smallest positive solutions to the equation 3sin(pi/4*x) = 2 are approximately:
x ≈ 0.9250, 7.1989, 13.4729, 19.7468.
3sin(pi/4*x)=2
sin(pi/4*x)=2/3
I am using radians, since you are using π in your angles.
π/4 x = .7297 or pi/4*x = 2.411895 because the sine is positive in quads I and II
x = .9291 or x = 3.07088
the period sin π/4*x is 2π/(π/4) = 8
so adding 8 to any existing angle will yield another answer.
so other solutions are:
So the 4 smallest positive solutions are:
x = .9291, 3.07088, 8..9291 and 10.07088