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What is a Periodic Function (Trigonometric Definition)?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
A periodic function is a function that repeats its values in regular intervals or periods. In trigonometry, this means the graph of the function looks exactly the same after a certain fixed horizontal distance, which is called its period.
Simple Example
Quick Example
Imagine a clock's minute hand. It starts at 12, moves around, and comes back to 12 every 60 minutes. The position of the minute hand is a periodic function of time, repeating every 60 minutes. Here, 60 minutes is the 'period'.
Worked Example
Step-by-Step
Let's check if f(x) = sin(x) is a periodic function with a period of 2*pi.
STEP 1: Understand the definition. For a function f(x) to be periodic with period P, f(x + P) must equal f(x).
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STEP 2: For f(x) = sin(x), we need to check if sin(x + 2*pi) = sin(x).
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STEP 3: Recall the trigonometric identity for sine: sin(A + B) = sin(A)cos(B) + cos(A)sin(B).
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STEP 4: Apply the identity: sin(x + 2*pi) = sin(x)cos(2*pi) + cos(x)sin(2*pi).
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STEP 5: We know the values of cos(2*pi) and sin(2*pi). cos(2*pi) = 1 and sin(2*pi) = 0.
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STEP 6: Substitute these values: sin(x + 2*pi) = sin(x)*(1) + cos(x)*(0).
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STEP 7: Simplify: sin(x + 2*pi) = sin(x) + 0 = sin(x).
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STEP 8: Since sin(x + 2*pi) = sin(x), f(x) = sin(x) is a periodic function with a period of 2*pi.
Answer: Yes, f(x) = sin(x) is a periodic function with a period of 2*pi.
Why It Matters
Periodic functions are super important in understanding waves – like sound waves in your phone or light waves from the sun, crucial in Physics. Engineers use them to design signals for mobile networks and for space communication (ISRO!). Doctors even use them to understand heartbeats and brain waves in Medicine.
Common Mistakes
MISTAKE: Thinking any repeating pattern is a periodic function without checking the exact values. | CORRECTION: A function is periodic only if f(x + P) = f(x) for ALL x in its domain, not just for a few points.
MISTAKE: Confusing the period with the frequency. | CORRECTION: The period is the length of one complete cycle (how long it takes to repeat), while frequency is how many cycles occur in a given unit of time.
MISTAKE: Assuming the period is always 2*pi for all trigonometric functions. | CORRECTION: While sin(x) and cos(x) have a period of 2*pi, functions like tan(x) have a period of pi, and functions like sin(2x) have a different period (pi).
Practice Questions
Try It Yourself
QUESTION: If a function f(x) has a period of 5, what is f(10) if f(0) = 7? | ANSWER: 7
QUESTION: Is the function f(x) = cos(x) a periodic function? If yes, what is its period? | ANSWER: Yes, it is periodic with a period of 2*pi.
QUESTION: A sound wave repeats its pattern every 0.002 seconds. What is the period of this sound wave? If a function describes this wave, what property would it have? | ANSWER: The period is 0.002 seconds. The function would be a periodic function.
MCQ
Quick Quiz
Which of the following is the defining property of a periodic function f(x) with period P?
f(x) = P
f(x + P) = f(x)
f(P) = x
f(x) + P = f(x)
The Correct Answer Is:
B
A periodic function repeats its values. So, if you add the period P to x, the function's value remains the same, i.e., f(x + P) = f(x).
Real World Connection
In the Real World
Think about the AC electricity supply in your home. The voltage goes up and down in a repeating pattern 50 times per second. This is a periodic function! Engineers at electricity boards use this concept to ensure stable power delivery across India.
Key Vocabulary
Key Terms
PERIOD: The smallest positive value P for which a function f(x) repeats itself, i.e., f(x + P) = f(x) | CYCLE: One complete repetition of a periodic pattern | TRIGONOMETRIC FUNCTION: Functions like sine, cosine, tangent that relate angles of a right-angled triangle to ratios of its sides, and show periodic behavior.
What's Next
What to Learn Next
Now that you understand periodic functions, you're ready to explore the graphs of specific trigonometric functions like sine, cosine, and tangent. You'll see how their periodic nature creates beautiful repeating wave patterns!
