Calculate values for Theta, given that 0* <= theta < 360*, and 4 Cos(theta) = 2
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![IMG20230610203859[1].jpg](https://www.freemathhelp.com/forum/data/attachments/33/33262-b0ae125446036d83eb59a71b79a9adfa.jpg "IMG20230610203859[1].jpg")
Dr.Peterson
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First, how did you get 1 degree? That makes no sense, as you can even see from your graph. Possibly you used your calculator incorrectly.
Second, the inverse cosine gives only one solution. The others can be derived from that by thinking about the definition of cosine (I like to use the unit-circle definition). What other angles will have the same cosine? This can also be seen from the symmetry of the graph.
Give it some thought (and check your textbook for ideas!), then write back.
Dr.Peterson
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I looked at it again, and have a guess as to what you may have done.
If you have the calculator set to degrees, and calculate [imath]\left(\cos\left(\frac{1}{2}\right)\right)^{-1}[/imath] instead of [imath]\cos^{-1}\left(\frac{1}{2}\right)[/imath], you will get 1.000038078, which you might be calling 1 degree.
If you did this, what you got is the reciprocal of the cosine of 1/2 degree, which is not at all what you want. The inverse cosine is the angle whose cosine is 1/2, which you find using the [imath]\cos^{-1}[/imath] button, not the [imath]\cos[/imath] button followed by the [imath]x^{-1}[/imath] button.
Or maybe you did something else entirely.
In any case, to find the other angle with the same cosine, you subtract what you got from 360. Can you see why?
If you have the calculator set to degrees, and calculate [imath]\left(\cos\left(\frac{1}{2}\right)\right)^{-1}[/imath] instead of [imath]\cos^{-1}\left(\frac{1}{2}\right)[/imath], you will get 1.000038078, which you might be calling 1 degree.
If you did this, what you got is the reciprocal of the cosine of 1/2 degree, which is not at all what you want. The inverse cosine is the angle whose cosine is 1/2, which you find using the [imath]\cos^{-1}[/imath] button, not the [imath]\cos[/imath] button followed by the [imath]x^{-1}[/imath] button.
Or maybe you did something else entirely.
In any case, to find the other angle with the same cosine, you subtract what you got from 360. Can you see why?
Good morning,
well, no, I used the cos−1 button..The sum as I did it is on my attachment, shows that I took the given expression, manipulated it to get the angle...which calculated to 1 deg....so, will the other angle be 359 deg?
blamocur
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What do you get if you take cosine of 1 degree?
Dr.Peterson
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It is correct to find [imath]\cos^{-1}\frac{1}{2}[/imath]; but the answer is not 1.
Please show us exactly what you did to get the answer of 1: What buttons did you push, and what kind of calculator do you have? An image of the calculator showing the answer (and anything your calculator displays that might indicate what it has done) may help.
The Highlander
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Please look at my 'alterations'/comments on your picture...
I have (tried to) replicate the (very helpful) red line that @Dr.Peterson already added to your graph at Post #3 (above) which shows that there is, indeed, more than one angle (θ) that has cos (θ) = ½ [0° ≤ θ ≤ 360°] (because that red line crosses your cosine wave at two places!).
We cannot help you to correct your mistaken final answer (that θ =1°) until you provide a more detailed explanation of exactly how you arrived at it.
You claim that you did use the "cos-1" button but what did you actually put into your calculator and what result was displayed on it?
Did you enter: cos-1 (0.5) or cos-1 (½) and get "1.047197551" displayed as the result (which you then rounded to "1")???
If that's what happened then your calculator has (inadvertently?) been changed to Radians! If you change it back to Degrees then you should get the correct answer (which is a whole number that requires no further rounding). ?
Please get back to us with your response.
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Oh my goodness!!!!..now I get 60deg...Why??...okay, uhm, ..honestly, I do not know...REALLY..I am stumped!!...Now I fully understand the whole thing....Once again, a BIG thank you..
The Highlander
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Good! But 60° is not the final answer.
You need to list all the angles that satisfy the equation.
So......?
The Highlander
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Why??
Because you are now doing it correctly! ?
The two most likely explanations (I can think of) for why you were getting it wrong are:-
1. Your calculator was (inadvertently?) set to Radian measure, then: [imath]cos^{-1} \left(\frac{1}{2}\right) \approx 1.04720[/imath]
or
2. You forgot* to press the "Alt" or "2ndF" button on you calculator, then: [imath]cos \left(\frac{1}{2}\right) \approx 0.99996[/imath]
(to get the "cos-1" function instead of just the "cos" function.)
Both of those 'answers' would round to 1 so do you remember whether your answer of θ = 1° was a rounded result (and can you recall what you rounded it from)?
Either of the above results ring any bells?
* forgot or just didn't press it hard enough.
NB: Getting 60° is only "half the battle"; we still need to see your complete answer (all poss. values of θ). ?
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Steven G
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Please look at YOUR graph from post#1.
You listed values 0 and 90. Now approximate where 1 should be (it should be very close to 0). Now that you are at 1 look up to the graph. Do you really think that the y value is 1/2? It seem from the graph that cos(0)=1. Well cos(1) will be very close to 1 and not near 1/2.
You listed values 0 and 90. Now approximate where 1 should be (it should be very close to 0). Now that you are at 1 look up to the graph. Do you really think that the y value is 1/2? It seem from the graph that cos(0)=1. Well cos(1) will be very close to 1 and not near 1/2.
good day Steven G,
well, the answer I remember VERY clearly was not rounded, the calculator actually gave me "1"...and not just once...a few times..I really cannot even remotely comprehend why that happened..Anycase, the other answer is 300 deg......
..Thank you kindly..
Steven G
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, you got exactly 1. The point I am making is that according to your graph there are only two places where the cos equals 1.Those two points are at 0 and 360. You need to be able to read a graph.good day Steven G,
well, the answer I remember VERY clearly was not rounded, the calculator actually gave me "1"...and not just once...a few times..I really cannot even remotely comprehend why that happened..Anycase, the other answer is 300 deg......
The Highlander
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Yes, you now have both of the correct answers: 60° & 300°. Well done! ?good day Steven G,
well, the answer I remember VERY clearly was not rounded, the calculator actually gave me "1"...and not just once...a few times..I really cannot even remotely comprehend why that happened..Anycase, the other answer is 300 deg......
If, as you say, your calculator was returning the whole number 1, then I cannot think of any explanation for what was going on there; I'm afraid it may just have to remain a mystery until the end of time. ?
However, I'm glad that you have managed to sort out your 'difficulties' and pleased that we may have been of some assistance to you along the way. ?
I wish you the best of luck with your future studies. ?