Graph one complete cycle of
y + cos x cos pi over6 + sin x sin pi/6 by first rewriting the right side in the form cos(A-B)
You probably meant to write
y = cos x cos pi over6 + sin x sin pi/6 which is
y = cos(x - pi/6)
first sketch very faintly, perhaps as a dotted line, the function y = cosx from 0 to 2pi
then move this graph pi/6 units to the right.
To rewrite the right side of the equation in the form cos(A-B), we can use the formula for the cosine of a difference of angles:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
Let's apply this formula to the given equation:
y + cos(x)cos(π/6) + sin(x)sin(π/6)
Using the formula, we have:
y + cos(x)cos(π/6) + sin(x)sin(π/6) = cos(x)cos(π/6) + sin(x)sin(π/6) + y
The right side of the equation is now in the form cos(A - B), where A = x and B = π/6.
Now, let's graph one complete cycle of the function by varying x from 0 to 2π.
Step 1: Choose values for x and calculate the corresponding value of the function.
For x = 0:
y = cos(0)cos(π/6) + sin(0)sin(π/6) + y
y = cos(0)cos(π/6) + 0
y = cos(0)cos(π/6)
y = 1 * cos(π/6)
y = cos(π/6)
y = √3/2
Therefore, when x = 0, y = √3/2.
Step 2: Repeat step 1 for different values of x within the range 0 to 2π.
For x = π/6:
y = cos(π/6)cos(π/6) + sin(π/6)sin(π/6) + y
y = cos(π/6)cos(π/6) + sin(π/6)sin(π/6) + √3/2
y = cos^2(π/6) + sin^2(π/6) + √3/2
y = 1 + √3/2
y = 2√3/2 + √3/2
y = 3√3/2
Therefore, when x = π/6, y = 3√3/2.
Repeat the same process for other values of x as follows:
x = π/3: y = √3/2
x = π/2: y = 1/2
x = 2π/3: y = 3/2
x = 5π/6: y = √3/2
x = π: y = 0
x = 7π/6: y = -√3/2
x = 4π/3: y = -√3/2
x = 3π/2: y = -1/2
x = 5π/3: y = -3/2
x = 11π/6: y = -√3/2
x = 2π: y = 0
You can plot these points on a graph and connect them to form one complete cycle of the function y = cos(x - π/6).
To rewrite the right side of the equation in the form cos(A - B), we can use the trigonometric identity:
cos(A - B) = cos A cos B + sin A sin B
Let's rewrite the right side of the equation using this identity:
y + cos x cos(pi/6) + sin x sin(pi/6)
= y + cos x (cos(pi/6) cos(pi/6)) + sin x (sin(pi/6) cos(pi/6))
= y + cos x cos^2(pi/6) + sin x sin(pi/6) cos(pi/6)
Now, let's simplify this expression further:
cos(pi/6) = sqrt(3)/2
sin(pi/6) = 1/2
Substituting these values back into our equation:
y + cos x (sqrt(3)/2)^2 + sin x (1/2)(sqrt(3)/2)
= y + cos x (3/4) + sin x (sqrt(3)/4)
Now we have rewritten the right side of the equation in the form cos(A - B).
To graph one complete cycle of this equation, we need to vary x from 0 to 2π (or 0° to 360°) and observe the corresponding values of y.
Here's how you can graph this equation:
1. Choose a range for x, such as 0 to 2π (or 0° to 360°).
2. Divide the chosen range into several points. For example, you can choose increments of π/4 or 45°.
3. Plug each value of x into the equation and calculate the corresponding value of y.
4. Plot the points (x, y) on a graph.
5. Connect the plotted points to form a smooth curve.
By following these steps, you will be able to graph one complete cycle of the given equation.