alicejanes
Badges: 6
Rep:
?
#1
Report Thread starter 7 months ago
#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?
0
reply
mathstutor24
Badges: 9
Rep:
?
#2
Report 7 months ago
#2
Think about how you would use your answer to part a) in the equation stated in part b)
0
reply
alicejanes
Badges: 6
Rep:
?
#3
Report Thread starter 7 months ago
#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.
0
reply
mathstutor24
Badges: 9
Rep:
?
#4
Report 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
0
reply
alicejanes
Badges: 6
Rep:
?
#5
Report Thread starter 7 months ago
#5
(Original post by mathstutor24)
Think about the range of y=cos(x)
I'm sorry, I dont understand
0
reply
mathstutor24
Badges: 9
Rep:
?
#6
Report 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?

Name:  Annotation 2020-06-08 171015.png
Views: 11
Size:  17.8 KB
0
reply
alicejanes
Badges: 6
Rep:
?
#7
Report Thread starter 7 months ago
#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?

Name:  Annotation 2020-06-08 171015.png
Views: 11
Size:  17.8 KB
so 1 and -1, would I make 1/8cos^4x+4 equal to each of these values and then solve ?
0
reply
mathstutor24
Badges: 9
Rep:
?
#8
Report 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 cos^{4}x they will both be 1, so setting cos(x) to 1 and -1 will give you the same answer. HINT: consider cos(x)=0
0
reply
davros
  • Study Helper
Badges: 16
Rep:
?
#9
Report 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
0
reply
mathstutor24
Badges: 9
Rep:
?
#10
Report 7 months ago
#10
if cos(x)=1, then cos^{4}(x)=1^{4}=1

Similarly, if cos(x)=-1, then cos^{4}(x)=(-1)^{4}=1
0
reply
mathstutor24
Badges: 9
Rep:
?
#11
Report 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
0
reply
davros
  • Study Helper
Badges: 16
Rep:
?
#12
Report 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
0
reply
mathstutor24
Badges: 9
Rep:
?
#13
Report 7 months ago
#13
Name:  Annotation 2020-06-08 171325.png
Views: 21
Size:  5.1 KB

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).
0
reply
alicejanes
Badges: 6
Rep:
?
#14
Report Thread starter 7 months ago
#14
(Original post by mathstutor24)
if cos(x)=1, then cos^{4}(x)=1^{4}=1

Similarly, if cos(x)=-1, then cos^{4}(x)=(-1)^{4}=1
so:
1/(8x1)+4 = 1/12 would be one of the values and the other would be 1/(8x0)+4 = 1/4 ??
0
reply
mathstutor24
Badges: 9
Rep:
?
#15
Report 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.
1
reply
alicejanes
Badges: 6
Rep:
?
#16
Report Thread starter 7 months ago
#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
1
reply
X

Quick Reply

Attached files
Write a reply...
Reply
new posts
Back
to top

If you don't put your camera on in online lessons, why is that?

My teacher doesn't want us to (20)
15.27%
No one else does (48)
36.64%
I'm embarrassed about my background (15)
11.45%
I feel self-conscious showing my face (40)
30.53%
We don't use a video platform (2)
1.53%
I don't have a camera (2)
1.53%
Something else (tell us in the thread) (4)
3.05%

Watched Threads

View All
Latest
My Feed

Oops, nobody has posted
in the last few hours.

Why not re-start the conversation?

Start new discussion

See more of what you like on
The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

Personalise