Solve cos(4x)-cos(2x)=0 ∀ 0<=x<=2pi ..............(0)
Normal solution:
1. use the double angle formula to decompose, and recall cos^2(x)+sin^2(x)=1
cos(4x)=cos^2(2x)-sin^2(2x)=2cos^2(2x)-1 .................(1)
2. substitute (1) in (0)
2cos^2(2x)-1-cos(2x)=0
3. substitute u=cos(2x)
2u^2-u-1=0
4. Solve for x
factor
(u-1)(u+1/2)=0
=> u=1 or u=-1/2
However, since cos(x) is an even function, so solutions to
{cos(2x)=1, cos(-2x)=1, cos(2x)=-1/2 and cos(-2x)} ...........(2)
are all solutions.
5. The cosine function is symmetrical about pi, therefore
cos(-2x)=cos(2*pi-2x),
solution (2) above becomes
{cos(2x)=1, cos(2pi-2x)=1, cos(2x)=-1/2, cos(2pi-2x)=-1/2}
6. Solve each case
cos(2x)=1 => x=0
cos(2pi-2x)=1 => cos(2pi-0)=1 => x=pi
cos(2x)=-1/2 => 2x=2pi/3 or 2x=4pi/3 => x=pi/3 or 2pi/3
cos(2pi-2x)=-1/2 => 2pi-2x=2pi/3 or 2pi-2x=4pi/3 => x=2pi/3 or x=4pi/3
Summing up,
x={0,pi/3, 2pi/3, pi, 4pi/3}
Answer:
use this
Step-by-step explanation:
velocity = distance / time…. So distance = velocity x time
Answer:
"After 64 seconds, ABOUT HALF of the brands of milk chocolate are LIKELY to melt and NONE of the brands of Dark Chocolate are likely to melt"
Step-by-step explanation:
The right answer is
"After 64 seconds, ABOUT HALF of the brands of milk chocolate are LIKELY to melt and NONE of the brands of Dark Chocolate are likely to melt"
We can easily check this solution by seeing the time it took for each chocolate to melt in the attached picture.
After 64 seconds have passed, half of the brands of milk chocolate had already melted, but the dark chocolate time starts after 200s
Answer: B. 35
Step-by-step explanation: The ratio of apples to oranges is 2:5. That means for every 2 apples there will be 5 oranges.
We want to know how many total pieces of fruit will be in the basket where there are 10 apples.
Lets use the ratio to get the total number of fruits when we have 10 apples.
2:5 there is 2 apples when there is 5 oranges;
4:10 there is 4 apples when there is 10 oranges;
6:15 there is 6 apples when there is 15 oranges;
8:20 there is 8 apples when there is 20 oranges;
10:25 there is 10 apples when there is 25 oranges;
Now simply add the total amount of apples and oranges. 10 + 25 = 35 total fruit.
The answer is 11 becauze the pattern works like this - 18÷2=9 20÷2=10 22÷2=11 Basically the X axis being divided by 2 which equals the Y axis (eg: X÷2=Y
