Solve 4sin(pi/4x) = 3 for the four smallest positive solutions
sin(pi/4 x) = 3/4
so π/4x = .848 or π-.848
π/4x = .848 or πx/4 = 2.2935
x = 1.08 or x = 2.902
but the period of sin (πx/4) is 8, so
x = 1.08 + 8k or x = 2.902 + 8k , where k is an integer
4 smallest positive solutions:
x = 1.08, 2.902, 9.08, and 10.902
Verification:
https://www.wolframalpha.com/input/?i=y+%3D+3%2C+y+%3D+4sin%28%CF%80x%2F4%29+from+0+to+15+
Why did the sine go to the beach? Because it had so many waves it just couldn't resist! But let's get serious now and solve this equation.
To solve 4sin(pi/4x) = 3, we need to isolate the sin(pi/4x) term and then take the inverse sin (or arcsin) of both sides.
Dividing both sides by 4 gives us:
sin(pi/4x) = 3/4
Now, we want to take the inverse sin of both sides. But before that, remember that the inverse sin has a range of -pi/2 to pi/2, so we need to limit our x values to make sure we only get the smallest positive solutions.
Taking the inverse sin of both sides, we have:
pi/4x = arcsin(3/4)
Using a calculator, we find that arcsin(3/4) ≈ 0.848 radians.
Now, we solve for x by multiplying both sides by 4/pi and simplifying:
x = (4/pi) * arcsin(3/4)
Now, we can find the four smallest positive solutions by substituting different values of n into the equation:
x₁ = (4/pi) * arcsin(3/4) ≈ 0.848 radians
x₂ = (4/pi) * (arcsin(3/4) + 2π) ≈ 4.989 radians
x₃ = (4/pi) * (arcsin(3/4) + 4π) ≈ 9.130 radians
x₄ = (4/pi) * (arcsin(3/4) + 6π) ≈ 13.271 radians
So the four smallest positive solutions to the equation 4sin(pi/4x) = 3 are approximately 0.848 radians, 4.989 radians, 9.130 radians, and 13.271 radians.
To solve the equation 4sin(pi/4x) = 3 for the four smallest positive solutions, we follow these steps:
Step 1: Divide both sides of the equation by 4:
sin(pi/4x) = 3/4
Step 2: Take the inverse sine (sin^(-1)) of both sides to isolate x:
pi/4x = sin^(-1)(3/4)
Step 3: Solve for x by multiplying both sides by 4 and dividing by pi:
x = (4/pi) * sin^(-1)(3/4)
Step 4: Calculate the four smallest positive solutions:
For the first solution, plug in n = 0 into the equation x = (4/pi) * sin^(-1)(3/4):
x₁ = (4/pi) * sin^(-1)(3/4)
For the second solution, plug in n = 1 into the equation:
x₂ = (4/pi) * sin^(-1)(3/4) + (2π/4)
For the third solution, plug in n = 2:
x₃ = (4/pi) * sin^(-1)(3/4) + (4π/4)
For the fourth solution, plug in n = 3:
x₄ = (4/pi) * sin^(-1)(3/4) + (6π/4)
These are the four smallest positive solutions for the equation 4sin(pi/4x) = 3.
To solve the equation 4sin(pi/4x) = 3 for the four smallest positive solutions, we need to isolate x. Here's how to do it:
First, divide both sides of the equation by 4 to obtain sin(pi/4x) = 3/4.
Next, we need to solve for pi/4x. To do that, we need to find the inverse sine (also known as arcsine) of both sides.
Take the inverse sine of sin(pi/4x) = 3/4:
arcsin(sin(pi/4x)) = arcsin(3/4).
Remember that the inverse sine function has a range of -pi/2 to pi/2. However, since we are looking for positive solutions, the range becomes 0 to pi/2.
Now, we have:
pi/4x = arcsin(3/4).
To find the four smallest positive solutions, we will solve for x in the range from 0 to pi/2.
Solution 1:
x = (4/pi) * arcsin(3/4).
To find the remaining solutions, we need to consider the periodic nature of the sine function. The sine function repeats itself every 2pi. So, we add multiples of 2pi to the equation for additional solutions.
Solution 2:
x = (4/pi) * (arcsin(3/4) + 2pi).
Solution 3:
x = (4/pi) * (arcsin(3/4) + 4pi).
Solution 4:
x = (4/pi) * (arcsin(3/4) + 6pi).
These are the four smallest positive solutions to the equation 4sin(pi/4x) = 3.
4 sin(pi/4 x) = 3
sin(pi/4 x) = 3/4
pi/4 x = 0.848 + k*2π
x = 1.08 + k/2
so, x = 1.08, 1.58, 2.08, 2.58