Here is correct question:
Solve
sin (pi/4*(x-6))=0.5, 0<x<360
Well, well, well, let's solve this equation and have some fun with math, shall we?
Now, to solve the equation sin(pi/4*(x-6)) = 0.5, we need to find the values of x that make this equation true within the given range of 0 < x < 360.
To do that, we'll first find the reference angle whose sine value is 0.5. Now, for all you non-math enthusiasts out there, a reference angle is the positive acute angle between the terminal side of an angle in standard position and the x-axis.
The sine value of 0.5 corresponds to an angle of 30 degrees or π/6 radians. So, let's add the reference angle to the original equation and solve for x.
pi/4*(x-6) = pi/6
Now, let's get rid of that pesky pi/4 by multiplying every term by 4/pi:
(x-6) = (4/pi) * pi/6
Simplifying further:
x - 6 = 4/6
And finally:
x = 6 + 2/3
So, within the given range of 0 < x < 360, the solution to the equation sin(pi/4*(x-6)) = 0.5 is x = 6 + 2/3.
I hope that brought a smile to your face as you dived into the world of trigonometry! If you have any more questions or need a good laugh, just let me know!
To solve the equation sin(pi/4*(x-6)) = 0.5, where 0 < x < 360, we can start by finding the general solution for sin(theta) = 0.5.
We know that sin(theta) = 0.5 is true for two values of theta: 30 degrees (or pi/6 radians) and 150 degrees (or 5*pi/6 radians).
Therefore, we can write the equation as:
pi/4*(x-6) = pi/6 + 2k*pi, (Equation 1)
pi/4*(x-6) = 5*pi/6 + 2k*pi, (Equation 2)
where k is an integer.
To solve Equation 1, we can start by isolating x:
x - 6 = (pi/6 + 2k*pi)*4/pi
x - 6 = (2/3 + 8k/3)*4
x - 6 = (8 + 32k)/3
x = (8 + 32k)/3 + 6
x = (8 + 32k + 18)/3
x = (26 + 32k)/3
Since 0 < x < 360, we can set up the inequality:
0 < (26 + 32k)/3 < 360
Simplifying the inequality, we get:
0 < 26 + 32k < 1080
Next, we can solve the inequality:
0 < 26 + 32k
Subtracting 26 from all parts of the inequality:
-26 < 32k
Dividing all parts of the inequality by 32 (keeping in mind that k is an integer):
-26/32 < k
-13/16 < k
The first part of the inequality tells us that k must be greater than -13/16. However, since k is an integer, the smallest value that satisfies this inequality is k = -1.
Next, we solve the second part of the inequality:
26 + 32k < 1080
Subtracting 26 from all parts of the inequality:
32k < 1054
Dividing all parts of the inequality by 32:
k < 1054/32
k < 65.9375
Since k is an integer, the largest value that satisfies this inequality is k = 65.
Therefore, the values of k that satisfy the inequality are -1 ≤ k ≤ 65.
Now, let's substitute the values of k back into Equation 1 to find the corresponding values of x:
For k = -1:
x = (26 + 32(-1))/3
x = (26 - 32)/3
x = -6/3
x = -2 (Not within the range of 0 < x < 360)
For k = 0:
x = (26 + 32(0))/3
x = 26/3
For k = 1:
x = (26 + 32(1))/3
x = (26 + 32)/3
x = 58/3
...
(Continue substituting the other values of k until k = 65)
For k = 65:
x = (26 + 32(65))/3
x = (26 + 2080)/3
x = 2106/3
x = 702
Therefore, the solutions to the equation sin(pi/4*(x-6)) = 0.5, where 0 < x < 360, are:
x = 26/3, 58/3, 90, 122, 154, 186, ..., 702
To solve the equation sin(pi/4*(x-6)) = 0.5, where 0 < x < 360, you can use the inverse sine function or the unit circle. I will explain both methods:
Method 1: Inverse Sine Function
Step 1: Take the inverse sine (arcsin) of both sides of the equation. This will remove the sine function.
arcsin(sin(pi/4*(x-6))) = arcsin(0.5)
Step 2: Simplify the left side using the property arcsin(sin(x)) = x if -pi/2 <= x <= pi/2.
Since 0 < x < 360, you need to find the values of x that satisfy this property.
pi/4*(x-6) = pi/6 + 2*pi*n, where n is an integer.
Step 3: Solve for x.
pi/4*(x-6) = pi/6 + 2*pi*n
Multiply both sides by 4/pi:
x - 6 = 4/6 + 8n
x = 2/3 + 8n + 6
x = 4/3 + 8n
So, the solutions for 0 < x < 360 are:
x = 4/3 + 8n, where n is an integer.
Method 2: Unit Circle
Step 1: Convert the equation sin(pi/4*(x-6)) = 0.5 to an angle.
sin(theta) = 0.5
The unit circle tells us that sin(theta) = 0.5 at two angles: 30 degrees and 150 degrees (or pi/6 and 5*pi/6 in radians).
Step 2: Set up the equation pi/4*(x-6) = theta and solve for x.
For theta = pi/6:
pi/4*(x-6) = pi/6
Multiply both sides by 4/pi:
x - 6 = 4/6
Simplify:
x - 6 = 2/3
x = 2/3 + 6
For theta = 5*pi/6:
pi/4*(x-6) = 5*pi/6
Multiply both sides by 4/pi:
x - 6 = 20/6
Simplify:
x - 6 = 10/3
x = 10/3 + 6
So, the solutions for 0 < x < 360 are:
x = 2/3 + 6 and x = 10/3 + 6.
I know that sin (π/6) or sin 30° = .5
so
(π/4)(x-6) = π/6 or (π/4)(x-6) = π - π/6 = 5π/6 (in II)
case 1
times 12 and divide by π
3(x-6) = 2
3x - 18 = 2
3x=20
x = 20/3
case 2
times 12 , and divide by π
3(x-6) = 10
3x - 18=10
3x = 28
x = 28/3