How do you solve cos(α+θ)=0 for different values of α?

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To solve the equation cos(α+θ)=0, the values of θ can be expressed as θ=π(n+1/2)−α, where n is an integer. When α is specified, such as α=3, specific solutions can be derived, but without a fixed α, the solutions become more complex. The inverse cosine function, cos^(-1)(y), is relevant here as it helps determine the angles corresponding to specific cosine values, but its domain and range must be considered. The discussion highlights the importance of understanding the periodic nature of the cosine function, as multiple θ values can satisfy the equation for a given α. Ultimately, the relationship between θ and α is crucial for finding valid solutions within the specified range of [0, 2π).
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Homework Statement
How do I solve cos(a+θ)=0?
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Hi,

I was trying to solve y=cos(α+θ) where α is a constant and θ has range [0,2π).

If α=3 is assumed, the solution is as shown below and it makes sense

\theta_{1}=\frac{3}{2}(\pi-2)
\theta_{2}=\frac{1}{2}(5 \pi-6)

But if I don't use any specific value for α, the solution doesn't make sense to me.

\pi n-\frac{5 \pi}{2} \leq \alpha \leq \pi n-\frac{\pi}{2} and \theta=\pi\left(n-\frac{1}{2}\right)-\alpha and n \in \mathbb{Z}

Source: https://www.wolframalpha.com/input/?i=solve+cos(α+θ)=0+for+θ+from+0+to+2pi

Somewhere I found the solution to be as shown below but don't understand why \cos ^{-1} y is used, and please note that α is not given any specific value:

\theta_{1}=\left(\cos ^{-1} \mathrm{y}\right)-\alpha
\theta_{2}=2 \pi-\left(\cos ^{-1} \mathrm{y}\right)-\alpha

Could you please help me? It's not homework but I still thought I better post it here nonetheless.
 
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\alpha+\theta=\pi(n+\frac{1}{2})
\theta=\pi(n+\frac{1}{2})-\alpha
integer n should be chosen properly so that
0 < \theta < 2\pi
You may make use of floor function to show solutions explicitly.
 
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PainterGuy said:
Somewhere I found the solution to be as shown below but don't understand why \cos ^{-1} y is used, and please note that α is not given any specific value:

\theta_{1}=\left(\cos ^{-1} \mathrm{y}\right)-\alpha
\theta_{2}=2 \pi-\left(\cos ^{-1} \mathrm{y}\right)-\alpha
What happens if you add ##\alpha## to those solutions for ##\theta##?

Do you know how the function ##\cos^{-1}## is defined?
 
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PeroK said:
What happens if you add ##\alpha## to those solutions for ##\theta##?

Do you know how the function ##\cos^{-1}## is defined?
Yes, I'd say I know how arccosine is defined. Cosine of an angle, cos(angle), gives you the ratio of base/hypotenuse and arccosine of the ratio, arccosine(base/hypotenuse), gives the corresponding angle.
 
PainterGuy said:
Yes, I'd say I know how arccosine is defined. Cosine of an angle, cos(angle), gives you the ratio of base/hypotenuse and arccosine of the ratio, arccosine(base/hypotenuse), gives the corresponding angle.
At an elementary level, the cosine is defined in terms of a right-angled triangle, but a question like this demands a more general definition.

More generally, ##\cos \theta## it is the x-coordinate on the unit circle corresponding to an angle ##\theta## as measured anti-clockwise from the positive x-axis.

You've missed the key point that the cosine is not a one-to-one function. E.g. ##\cos \theta = \cos (-\theta)##. It is not necessarily the case, therefore, that ##\cos^{-1}(cos \theta) = \theta##

This means you need a slightly more sophisticated definition of the inverse cosine. I'll let you look that up.
 
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PeroK said:
This means you need a slightly more sophisticated definition of the inverse cosine. I'll let you look that up.

Thank you for pointing this out.

On a unit circle the cosine is the length of the adjacent side (which is the x-coordinate) divided by the length of the hypotenuse (which is 1). So, as you said, the cosine is just the x-coordinate.

The domain of the inverse cosine function is [−1,1] and the range is [0,π]. That means a positive value will yield a 1st quadrant angle and a negative value will yield a 2nd quadrant angle.Helpful links:
https://www.mathwords.com/c/cosine_inverse.htm
/watch?v=ZEAgtXTIxk0 (add youtube.com in front)
/watch?v=xEGZsWgCn-Q
https://www.khanacademy.org/math/al...0c9fb89:unit-circle/a/trig-unit-circle-review
 
anuttarasammyak said:
\alpha+\theta=\pi(n+\frac{1}{2})
\theta=\pi(n+\frac{1}{2})-\alpha
integer n should be chosen properly so that
0 < \theta < 2\pi
You may make use of floor function to show solutions explicitly.

Thanks!

But it also involves α in addition to n; two variables.

Also, I don't see how the floor function could be used here.
 
PainterGuy said:
Thank you for pointing this out.

On a unit circle the cosine is the length of the adjacent side (which is the x-coordinate) divided by the length of the hypotenuse (which is 1). So, as you said, the cosine is just the x-coordinate.

The domain of the inverse cosine function is [−1,1] and the range is [0,π]. That means a positive value will yield a 1st quadrant angle and a negative value will yield a 2nd quadrant angle.Helpful links:
https://www.mathwords.com/c/cosine_inverse.htm
/watch?v=ZEAgtXTIxk0 (add youtube.com in front)
/watch?v=xEGZsWgCn-Q
https://www.khanacademy.org/math/al...0c9fb89:unit-circle/a/trig-unit-circle-review
Okay. So for every point ##y## in ##[-1,1]## there are two values of ##\theta## for which ##\cos \theta = y##.

Can you see the relationship between the two values of ##y##?
 
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PeroK said:
Can you see the relationship between the two values of ##y##?
I think both values of y should be equal.
 
  • #10
PainterGuy said:
Also, I don't see how the floor function could be used here.
Say
\frac{1}{4}-\frac{\alpha}{2\pi}=123456.789
\frac{3}{4}-\frac{\alpha}{2\pi}=123457.289
You will be able to reduce them to 0.789, 0.289 using floor function.

.
 
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  • #11
PainterGuy said:
I think both values of y should be equal.
They are not. The graph of ##\cos \theta## is symmetric about ##\theta = \pi##. For every value of ##y## there are two values of ##\theta## for which ##\cos \theta = y##: one in the interval ##[0, \pi]## and one in the interval ##[\pi, 2\pi]##. The only exception is ##y = -1##, for which there is only ##\theta = \pi##.

This symmetry may be expressed in two ways (these are common trig identities):$$\cos (\pi - \theta) = \cos (\pi + \theta)$$ or $$\cos \theta = \cos(2\pi - \theta)$$You should convince yourself of these by looking at the graph of ##\cos \theta## on the interval ##[0, 2\pi]##.
 
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  • #12
Whoah, this seems to have gone WAY off track. Assuming this is the question:
PainterGuy said:
How do I solve cos(a+θ)=0?
Then you are going about this completely the wrong way.

Here is the first step along the right way: for what values of ## x ## is ## \cos x = 0 ## true?
 
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  • #13
pbuk said:
Whoah, this seems to have gone WAY off track. Assuming this is the question:

Then you are going about this completely the wrong way.

Here is the first step along the right way: for what values of ## x ## is ## \cos x = 0 ## true?
If you read the OP that ##0## should be a general ##y##. Given the solution, I assume the first line is a typo.
 
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  • #14
PeroK said:
If you read the OP that ##0## should be a general ##y##. Given the solution, I assume the first line is a typo.
That is also a possibility, but it doesn't fit with this part of the OP which is clearly referring to ## \cos (\alpha + \theta) = 0 ##.
PainterGuy said:
If α=3 is assumed, the solution is as shown below and it makes sense

\theta_{1}=\frac{3}{2}(\pi-2)
\theta_{2}=\frac{1}{2}(5 \pi-6)

But if I don't use any specific value for α, the solution doesn't make sense to me.

\pi n-\frac{5 \pi}{2} \leq \alpha \leq \pi n-\frac{\pi}{2} and \theta=\pi\left(n-\frac{1}{2}\right)-\alpha and n \in \mathbb{Z}

Source: https://www.wolframalpha.com/input/?i=solve+cos(α+θ)=0+for+θ+from+0+to+2pi
 
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  • #15
@PainterGuy please clarify whether you are only looking at the special case where ##y = 0##. That said, this is as good an opportunity as any to understand the inverse cosine function generally!
 
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  • #16
PeroK said:
@PainterGuy please clarify whether you are only looking at the special case where ##y = 0##. That said, this is as good an opportunity as any to understand the inverse cosine function generally!
Yes, I was trying to find the roots of y=cos(α+θ) which would require y=0. α is a constant and I'd assume that it's not zero.
 
  • #17
PainterGuy said:
Yes, I was trying to find the roots of y=cos(α+θ) which would require y=0. α is a constant and I'd assume that it's not zero.
Okay, so we have an answer in post #2. Without knowing a range for ##\alpha## it looks a bit messy to try to keep ##\theta## in a certain range. Is that what you want to do?
 
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  • #18
PainterGuy said:
Yes, I was trying to find the roots of y=cos(α+θ) which would require y=0.
Why are you writing this in such a complicated way? You don't need to introduce ## y ##, you are simply trying to find values of ## \theta ## that satisfy ## cos (\alpha + \theta) = 0 ##.

The first step to answer that question is answering:
pbuk said:
for what values of ## x ## is ## \cos x = 0 ## true?
 
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  • #19
Note that to get the answer from Wolfram Alpha you simply need to enter "solve cos(\alpha + \theta) = 0 for \theta", but in order to use a tool like Wolfram Alpha you need to have some understanding of the basics.
 
  • #20
PeroK said:
Okay, so we have an answer in post #2. Without knowing a range for ##\alpha## it looks a bit messy to try to keep ##\theta## in a certain range. Is that what you want to do?
Yes, I was trying to keep the theta in the given range without knowing the range for alpha.
 
  • #21
pbuk said:
Why are you writing this in such a complicated way? You don't need to introduce ## y ##, you are simply trying to find values of ## \theta ## that satisfy ## cos (\alpha + \theta) = 0 ##.

The first step to answer that question is answering:

cos(x)=0 for x=pi/2 and x=3pi/2 if x is restricted to 0 to 2pi.
 
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  • #22
PainterGuy said:
cos(x)=0 for x=pi/2 and x=3pi/2 if x is restricted to 0 to 2pi.

Right, let's just look at the first solution, ## x = \dfrac \pi 2 ##.

Look back at the original problem ## \cos(\alpha + \theta) = 0 ##. What I have done is made a substitution ## x = \alpha + \theta ##. Using substitutions is an important tool in solving mathematical problems; choosing the right substitution makes the problem easier to solve. We can now write an equation with the substitution I made on the LHS and the solution you found for x on the RHS:
$$ \alpha + \theta = \dfrac \pi 2 $$
and we can simplify this to
$$ \theta = \dfrac \pi 2 - \alpha $$
Note that this will only give ## 0 \le \theta \lt 2 \pi ## for certain values of ## \alpha ## (what are these?)

You can do something similar with your other solution for ## x ##, and again look at how this works for different values of ## \alpha ##.
 
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