To solve for the period of a trigonometric function, you'll use different formulas depending on whether the function is sine/cosine or tangent/cotangent. The key is to identify the 'B' value in the function's equation.
Understanding the 'B' Value
The general form of a trigonometric function that involves a period is:
y = A trig(B x + C) + D
Where:
- A affects the amplitude.
- B affects the period.
- C affects the horizontal shift.
- D affects the vertical shift.
It is the 'B' that we are focused on when calculating period.
Formulas for Calculating Period
The formula for finding the period depends on the specific trigonometric function.
| Trigonometric Function | Formula for Period |
|---|---|
| Sine (sin) and Cosine (cos) | Period = 2π / |
| Tangent (tan) and Cotangent (cot) | Period = π / |
Note: |B| represents the absolute value of B.
Step-by-Step Approach:
- Identify the trigonometric function: Determine if you're working with sine, cosine, tangent, or cotangent.
- Identify the 'B' value: Locate the coefficient of 'x' within the trigonometric function.
- Apply the appropriate formula: Use the corresponding formula from the table above.
- Calculate: Divide 2π or π by the absolute value of B to determine the period.
Examples
-
Example 1: Find the period of y = 3sin(2x)
- Function: Sine
- B value: 2
- Period: 2π / |2| = π
-
Example 2: Find the period of y = cos(x/2)
- Function: Cosine
- B value: 1/2
- Period: 2π / |1/2| = 4π
-
Example 3: Find the period of y = 2tan(3x)
- Function: Tangent
- B value: 3
- Period: π / |3| = π/3
-
Example 4: Find the period of y = 4cot(x/4)
- Function: Cotangent
- B value: 1/4
- Period: π / |1/4| = 4π
Why Use Absolute Value?
The absolute value of B (|B|) is crucial because period is always a positive value, so the absolute value ensures that the final period is a positive value, regardless of whether B is positive or negative.
Conclusion
Solving for period involves identifying the trigonometric function and the value of 'B', then applying the correct formula to calculate the period which is:
- 2π / |B| for sine and cosine.
- π / |B| for tangent and cotangent.
