# A-level Trig Help

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#1
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?
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7 months ago
#2
Think about how you would use your answer to part a) in the equation stated in part b)
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#3
(Original post by mathstutor24)
Think about how you would use your answer to part a) in the equation stated in part b)
I have now got 1/(8cos^4x + 4) which I have written in terms of sec, 8sec^4x + 1/4. However there is no equals so i know i cannot move the 1/4 across, so I dont know where to go from this equation.
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7 months ago
#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?
Last edited by mathstutor24; 7 months ago
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#5
(Original post by mathstutor24)
Think about the range of y=cos(x)
I'm sorry, I dont understand
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7 months ago
#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?

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#7
(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?

so 1 and -1, would I make 1/8cos^4x+4 equal to each of these values and then solve ?
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7 months ago
#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
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7 months ago
#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.
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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7 months ago
#10
if , then

Similarly, if , then
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7 months ago
#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
That's what I'm trying to help them see, davros
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7 months ago
#12
(Original post by mathstutor24)
That's what I'm trying to help them see, davros
Yes I think we were typing at the same time - I was concerned they were going down the wrong route with that invalid fraction conversion and weren't following your excellent advice
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7 months ago
#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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#14
(Original post by mathstutor24)
if , then

Similarly, if , then
so:
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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7 months ago
#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.
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#16
(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.
thank you, this is really helpful
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