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Stuck On Trigonometry
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I have the equation 13Cos(x+22.6), and it's asking me to find when the maximum and minium occur. How would I do that?
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I still feel max/min involves differentiation but anyway yes the max value of a Cosine is 1. Are you splitting out your Cos(x+22.6) correctly. Again this is from fuzzy memory but I think Cos(A+B) is Cos²A - Sin²B (I haven't checked this). I won't say anymore because there's no point, at least 1 person will see this and give you a far more comprehensive and correct answer if you are patient.
Short answer: Set Cos(x+22.6) = 1 for maximum value and Cos(x+22.6) = -1 for minimum. Maximum occurs when (x+22.6) = 0, 360, 720 etc and rearrange; minimum occurs when (x+22.6) = 180, 540, 900 etc and rearrange.
* * * * * * *
Long answer:
What Prudie said -- although if, as I suspect, x is measured in degrees then differentiation becomes just a shade trickier than it needs to be. As I'll explain later this extra difficulty doesn't actually matter although it depends on the precise question being asked -- and, indeed, this is probably more work than is needed or required.
It is a fact that Cos(y) has a maximum when y = 0° (and also 360°, 720°, 1080° etc., and also -360°, -720° etc.) This maximum value equals 1. It's important to learn this property of cosine/ sine functions, and something you should do at least once or twice in your life is plot Sin(x) and Cos(x) (eg on Excel) between -360° and 360° and learn/ label all the properties of the curves.
Anyway, Cos(y), whatever the form of y, has a maximum value of 1 at y = 0. For our problem, we can replace y by what is given, which is y=x+22.6, and then solve x+22.6=0 to find x = -22.6° (and also x= -22.6° + 360°, etc., depending on how the question is worded).
Similarly, Cos(y) has a minimum value of -1, which occurs whenever y = 180°, 540°, 900° etc., as well as -180°, -540°, -900° etc. So to find the minimum, we can set:
x+22.6 = y
13Cos(x+22.6) = 13Cos(y)
Minimum value Cos(y) = -1
Minimum at y = 180°
Hence y= x+22.6 = 180°
Hence x = 180° - 22.6° = 157.4°
And for a complete solution we could add or subtract integer multiples of 360°.
Prudie's suggestion is indeed correct, the difference between that instead of learning the properties of Cosine we need instead to know the properties of Sine. When differentiating what we're given, which is 13Cos(x+22.6), we would end up with an answer 13 Sin(x+22.6)*(constant), where the constant is probably Pi/180 or 180/Pi. As it happens, it doesn't matter because all we are interested in when finding maximum and minimum points is when the gradient function, which is what you get when you differentiate, becomes 0, and Sin(y) = 0 when y = 0°, 180°, 360°, 540° etc and always adding (or subtracting) whole-number multiples of 180. These values are the same as we had before so again you'd just set x + 22.6 = 0° or x + 22.6 = 180° and the problem is the same. The remaining trick is knowing which is a maximum and which is a minimum; since that would require differentiating a second time, I'm guessing that this question is more about knowing the properties of Sin/ Cos than it is about differentiating.
* * * * * * *
Long answer:
What Prudie said -- although if, as I suspect, x is measured in degrees then differentiation becomes just a shade trickier than it needs to be. As I'll explain later this extra difficulty doesn't actually matter although it depends on the precise question being asked -- and, indeed, this is probably more work than is needed or required.
It is a fact that Cos(y) has a maximum when y = 0° (and also 360°, 720°, 1080° etc., and also -360°, -720° etc.) This maximum value equals 1. It's important to learn this property of cosine/ sine functions, and something you should do at least once or twice in your life is plot Sin(x) and Cos(x) (eg on Excel) between -360° and 360° and learn/ label all the properties of the curves.
Anyway, Cos(y), whatever the form of y, has a maximum value of 1 at y = 0. For our problem, we can replace y by what is given, which is y=x+22.6, and then solve x+22.6=0 to find x = -22.6° (and also x= -22.6° + 360°, etc., depending on how the question is worded).
Similarly, Cos(y) has a minimum value of -1, which occurs whenever y = 180°, 540°, 900° etc., as well as -180°, -540°, -900° etc. So to find the minimum, we can set:
x+22.6 = y
13Cos(x+22.6) = 13Cos(y)
Minimum value Cos(y) = -1
Minimum at y = 180°
Hence y= x+22.6 = 180°
Hence x = 180° - 22.6° = 157.4°
And for a complete solution we could add or subtract integer multiples of 360°.
Prudie's suggestion is indeed correct, the difference between that instead of learning the properties of Cosine we need instead to know the properties of Sine. When differentiating what we're given, which is 13Cos(x+22.6), we would end up with an answer 13 Sin(x+22.6)*(constant), where the constant is probably Pi/180 or 180/Pi. As it happens, it doesn't matter because all we are interested in when finding maximum and minimum points is when the gradient function, which is what you get when you differentiate, becomes 0, and Sin(y) = 0 when y = 0°, 180°, 360°, 540° etc and always adding (or subtracting) whole-number multiples of 180. These values are the same as we had before so again you'd just set x + 22.6 = 0° or x + 22.6 = 180° and the problem is the same. The remaining trick is knowing which is a maximum and which is a minimum; since that would require differentiating a second time, I'm guessing that this question is more about knowing the properties of Sin/ Cos than it is about differentiating.
Incidentally Prudie, Cos(A+B) = Cos(A)Cos(B) - Sin(A)Sin(B) -- I don't think that using this result helps here as it would give something like
0.92 Cos(x) - 0.38 Sin(x) = 1 ,
which doesn't look solvable to me without packing everything back together again.
* * * * *
I skipped some working above, I don't know if it's important. From the line
Cos(x+22.6) = 1
Strictly, you should take the inverse cosine function of both sides, here written ArcCos:
ArcCos (Cos(x+22.6)) = ArcCos(1)
By definition ArcCos and Cos cancel each other out on the left, while on the right any calculator tells you that ArcCos(1) = 0, and you can always add or subtract multiples of 360° to this. Hence:
(x+22.6) = ArcCos(1) = 0°, 360° etc.
And then rearrange for x.
0.92 Cos(x) - 0.38 Sin(x) = 1 ,
which doesn't look solvable to me without packing everything back together again.
* * * * *
I skipped some working above, I don't know if it's important. From the line
Cos(x+22.6) = 1
Strictly, you should take the inverse cosine function of both sides, here written ArcCos:
ArcCos (Cos(x+22.6)) = ArcCos(1)
By definition ArcCos and Cos cancel each other out on the left, while on the right any calculator tells you that ArcCos(1) = 0, and you can always add or subtract multiples of 360° to this. Hence:
(x+22.6) = ArcCos(1) = 0°, 360° etc.
And then rearrange for x.
Hi Jim thax for the answer
I used to loathe this sort of question....
really short answer is the max is gonna be when Cos X+22.6 = +1
and the min when it equals -1
and the maxima are +13 and the min -13 at x = -22.6, 337.4, etc
it strikes me that differentiating etc
means you wouldnt finish the paper ....
No one has said that this isnt an equation but a term
or a function .... ( no equals sign )
I used to loathe this sort of question....
really short answer is the max is gonna be when Cos X+22.6 = +1
and the min when it equals -1
and the maxima are +13 and the min -13 at x = -22.6, 337.4, etc
it strikes me that differentiating etc
means you wouldnt finish the paper ....
No one has said that this isnt an equation but a term
or a function .... ( no equals sign )
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