1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
alexandr1967 [171]
3 years ago
6

How to know if a function is periodic without graphing it ?

Mathematics
1 answer:
zhenek [66]3 years ago
4 0
A function f(t) is periodic if there is some constant k such that f(t+k)=f(k) for all t in the domain of f(t). Then k is the "period" of f(t).

Example:

If f(x)=\sin x, then we have \sin(x+2\pi)=\sin x\cos2\pi+\cos x\sin2\pi=\sin x, and so \sin x is periodic with period 2\pi.

It gets a bit more complicated for a function like yours. We're looking for k such that

\pi\sin\left(\dfrac\pi2(t+k)\right)+1.8\cos\left(\dfrac{7\pi}5(t+k)\right)=\pi\sin\dfrac{\pi t}2+1.8\cos\dfrac{7\pi t}5

Expanding on the left, you have

\pi\sin\dfrac{\pi t}2\cos\dfrac{k\pi}2+\pi\cos\dfrac{\pi t}2\sin\dfrac{k\pi}2

and

1.8\cos\dfrac{7\pi t}5\cos\dfrac{7k\pi}5-1.8\sin\dfrac{7\pi t}5\sin\dfrac{7k\pi}5

It follows that the following must be satisfied:

\begin{cases}\cos\dfrac{k\pi}2=1\\\\\sin\dfrac{k\pi}2=0\\\\\cos\dfrac{7k\pi}5=1\\\\\sin\dfrac{7k\pi}5=0\end{cases}

The first two equations are satisfied whenever k\in\{0,\pm4,\pm8,\ldots\}, or more generally, when k=4n and n\in\mathbb Z (i.e. any multiple of 4).

The second two are satisfied whenever k\in\left\{0,\pm\dfrac{10}7,\pm\dfrac{20}7,\ldots\right\}, and more generally when k=\dfrac{10n}7 with n\in\mathbb Z (any multiple of 10/7).

It then follows that all four equations will be satisfied whenever the two sets above intersect. This happens when k is any common multiple of 4 and 10/7. The least positive one would be 20, which means the period for your function is 20.

Let's verify:

\sin\left(\dfrac\pi2(t+20)\right)=\sin\dfrac{\pi t}2\underbrace{\cos10\pi}_1+\cos\dfrac{\pi t}2\underbrace{\sin10\pi}_0=\sin\dfrac{\pi t}2

\cos\left(\dfrac{7\pi}5(t+20)\right)=\cos\dfrac{7\pi t}5\underbrace{\cos28\pi}_1-\sin\dfrac{7\pi t}5\underbrace{\sin28\pi}_0=\cos\dfrac{7\pi t}5

More generally, it can be shown that

f(t)=\displaystyle\sum_{i=1}^n(a_i\sin(b_it)+c_i\cos(d_it))

is periodic with period \mbox{lcm}(b_1,\ldots,b_n,d_1,\ldots,d_n).
You might be interested in
To add or subtract fractions with different denominators, first find equivalent fractions with a _____ denominator.
frosja888 [35]
Common denominator is your answer.
7 0
2 years ago
Read 2 more answers
Which is the best stock you can invest now in 2021, or where I can learn to invest in stock market​
AlladinOne [14]
There really is no ‘best stock’ to invest in. But some good stocks to invest in are Amazon, Disney, and Apple. All are companies that have had tremendous growth and appear to still be growing. You can also go onto Yahoo finance, they will tell you whether they think the stock is over priced and you can also see the growth pattern over time.

Hope this helps! Please make me the brainliest, it’s not necessary but appreciated, I put a lot of effort and research into my answers. Have a good day, stay safe and stay healthy.
3 0
3 years ago
7 in.
fgiga [73]
258 cubic inches

Since the boxes are kind of on their own—>
First box: 7*2*12 = 168
Second box: 5*2*9=90
168+90 = 258
6 0
3 years ago
What is the surface area?
frosja888 [35]
98cm^2 - bigger triangles are 28 each, bottom one is 14. (28 x 3) + 14=98
3 0
3 years ago
Find the distance between the following points: (0,0,0) and (-2,6,3)
marusya05 [52]

9514 1404 393

Answer:

  3.  7

Step-by-step explanation:

The distance formula applies in 3 dimensions as well as 2.

  d = √((x2 -x1)² +(y2 -y1)² +(z2 -z1)²)

  d = √((-2)² +6² +3²) = √(4 +36 +9) = √49

  d = 7

The distance between the two points is 7 units.

6 0
2 years ago
Other questions:
  • Please help what is 1 plus 1 really hard please help
    5·2 answers
  • Please helppp I need help ASAP!
    9·1 answer
  • -3c = 36<br><br> Show work please
    9·1 answer
  • Using the fundamental poisson brackets find the values of a and b for which the equations represent a canonical transformation g
    13·1 answer
  • 7x 3/4 qeqeqeqeqeqeqeqeqeqe
    12·1 answer
  • Which regular polygon would have each of its interior angles measure 140°?
    6·1 answer
  • Hello, I give brainliest. Do not answer questions just to get points or you will get reported. Your answer may be incorrect but
    5·2 answers
  • Witch shape does it make​
    12·1 answer
  • It cost 60$ to for 4 apples how much would 60 apples cost
    13·2 answers
  • Find the distance between each pair of points (round answer to the nearest tenth if necessary)
    12·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!