You appear to understand this well. To answer your question about the units of the 'x' coordinate (which is actually just theta):
Your graph will show y vs. theta, so the units on the horizontal axis will be radians.
If you solve the equation 2 pi n + 9 theta - 3 pi = arccos(-4.345 / 7), the 3 pi and 2 pi n are in radians (any angular quantity not given in units is in radians as the default unit), and the value of the arccos is in radians. The only quantity not in radians is theta, so when you solve to get
theta = (2 pi n + arccos(-4.345 / 7)) / 9
the numerator of the right-hand side is in radians, the denominator is the unitless number 9, and the quotient is therefore in radians.
Sir,
Following up on this question 7cos( 9 `theta - 3 `pi) = -4.345 .
Had I graphed y1= 7cos( 9 `theta - 3 `pi) and y2=-4.345 and seen the points of intersection, I would have seen that graph had to make a full rotation (2pi) before it intersected the line y=-4.345..I would also have seen that it intersected the line on the way down and on the way up, but I would not have had the where-with-all to describe it in a function. So, Thank you for the clarification !!
I went back over your notes and seen your example of 2pi*n and it much clearer now.
If I project the points of intersection up to the x axis what values will they have. I’m having a time of it trying to determine the correct units..
Example:
(9theta-3pi)=cos^-1(-4.345/7) yields a value for theta of 1.296 radians but what is this value on the x axis..
Similarly , if I set (9theta-3pi)=cos^-1(-4.345/7)+ 2pi*n or (9theta-3pi)=2pi*n- cos^-1(-4.345/7) over domain 0 to 2pi , how can I confirm the value of the x coordinate . I’m back to the confusing issue of the value of angular freq compared to clock time (t) ..Sorry..
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