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It is given that f(x)=sin(x+30)+cos(x+60)
a) show that f(x)=cosx, hence show that f(4x)+4f(2x) = Acos^4x-B
b) determine the greatest and least values of 1/f(4x)+4f(2x)+7 as x varies
c) solve the equation, sin(12a+30)+cos(12a+60)+4sin(6a+ 30)+4cos(6a+60)=1 for 0<a<60
I have worked out part a as A=8 and B=3, which is correct, however I am struggling to find a starting point for both parts b and c, any ideas?
a) show that f(x)=cosx, hence show that f(4x)+4f(2x) = Acos^4x-B
b) determine the greatest and least values of 1/f(4x)+4f(2x)+7 as x varies
c) solve the equation, sin(12a+30)+cos(12a+60)+4sin(6a+ 30)+4cos(6a+60)=1 for 0<a<60
I have worked out part a as A=8 and B=3, which is correct, however I am struggling to find a starting point for both parts b and c, any ideas?
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#2
Think about how you would use your answer to part a) in the equation stated in part b)
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(Original post by mathstutor24)
Think about how you would use your answer to part a) in the equation stated in part b)
Think about how you would use your answer to part a) in the equation stated in part b)
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#4
Think about the range of y=cos(x)
i.e. what is the biggest and smallest value cos(x) can equal. How can you use this information to determine the greatest and least values?
i.e. what is the biggest and smallest value cos(x) can equal. How can you use this information to determine the greatest and least values?
Last edited by mathstutor24; 7 months ago
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(Original post by mathstutor24)
Think about the range of y=cos(x)
Think about the range of y=cos(x)
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#6
what is the biggest value that cos(x) can be, regardless of the value of x?
similarly, what is the smallest value that cos(x) can be?
![Name: Annotation 2020-06-08 171015.png
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similarly, what is the smallest value that cos(x) can be?
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(Original post by mathstutor24)
what is the biggest value that cos(x) can be, regardless of the value of x?
similarly, what is the smallest value that cos(x) can be?
![Name: Annotation 2020-06-08 171015.png
Views: 11
Size: 17.8 KB]()
what is the biggest value that cos(x) can be, regardless of the value of x?
similarly, what is the smallest value that cos(x) can be?
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#8
Almost! Yes, the largest and smallest values of cos(x) are 1 and -1. So... if you think about them in terms of 
they will both be 1, so setting cos(x) to 1 and -1 will give you the same answer. HINT: consider cos(x)=0
they will both be 1, so setting cos(x) to 1 and -1 will give you the same answer. HINT: consider cos(x)=0
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#9
(Original post by alicejanes)
I have now got 1/(8cos^4x + 4) which I have written in terms of sec, 8sec^4x + 1/4.
I have now got 1/(8cos^4x + 4) which I have written in terms of sec, 8sec^4x + 1/4.
Hint: you don't actually need to "convert" the fraction - you're just interested in how big or small the denominator can be so you can work out how small (or big) the overall fraction can be
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#11
(Original post by davros)
Are you sure those 2 expressions are equivalent?
Hint: you don't actually need to "convert" the fraction - you're just interested in how big or small the denominator can be so you can work out how small (or big) the overall fraction can be
Are you sure those 2 expressions are equivalent?
Hint: you don't actually need to "convert" the fraction - you're just interested in how big or small the denominator can be so you can work out how small (or big) the overall fraction can be
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#12
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#13
The bit that I've highlighted in blue - you should be able to figure out the max and min value that can equal using the info I've posted already. As davros said, you need to calculate the biggest and smallest denominator, which you can do from your knowledge of the range of cos(x).
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1/(8x1)+4 = 1/12 would be one of the values and the other would be 1/(8x0)+4 = 1/4 ??
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#15
Yes This kind of thing (where they ask you for a min or max value involving trig) comes up frequently in A Level maths. It's usually involving sine and cosine, so remember a tip is to remember your trig graphs - the max and min values that sin(x) and cos(x) can be - and figure it out from there. If your equation didn't include an even exponent, then you would have used the min value of cos(x) (i.e. -1) to get the least value, since (-1)^odd would result in a negative value.
This kind of thing (where they ask you for a min or max value involving trig) comes up frequently in A Level maths. It's usually involving sine and cosine, so remember a tip is to remember your trig graphs - the max and min values that sin(x) and cos(x) can be - and figure it out from there. If your equation didn't include an even exponent, then you would have used the min value of cos(x) (i.e. -1) to get the least value, since (-1)^odd would result in a negative value.
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(Original post by mathstutor24)
Yes This kind of thing (where they ask you for a min or max value involving trig) comes up frequently in A Level maths. It's usually involving sine and cosine, so remember a tip is to remember your trig graphs - the max and min values that sin(x) and cos(x) can be - and figure it out from there. If your equation didn't include an even exponent, then you would have used the min value of cos(x) (i.e. -1) to get the least value, since (-1)^odd would result in a negative value.
Yes
This kind of thing (where they ask you for a min or max value involving trig) comes up frequently in A Level maths. It's usually involving sine and cosine, so remember a tip is to remember your trig graphs - the max and min values that sin(x) and cos(x) can be - and figure it out from there. If your equation didn't include an even exponent, then you would have used the min value of cos(x) (i.e. -1) to get the least value, since (-1)^odd would result in a negative value.
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