Finding the Period of a Sine or Cosine Function
Find the coefficient B in your sine or cosine function. Your function should follow this function formula: f ( x ) = A s i n ( B x + C ) + D {\displaystyle f(x)=Asin(Bx+C)+D} {\displaystyle f(x)=Asin(Bx+C)+D} or f ( x ) = A c o s ( B x + C ) + D {\displaystyle f(x)=Acos(Bx+C)+D} {\displaystyle f(x)=Acos(Bx+C)+D}. Identify the value of B in your formula. None of the other coefficients will affect the period of the sine wave, so you can disregard them for now.
Plug B into the formula P e r i o d = 2 p i / | B | {\displaystyle Period=2pi/|B|} {\displaystyle Period=2pi/|B|}. This formula is used for both sine and cosine functions, so you can follow the same steps for either. Divide 2 p i {\displaystyle 2pi} {\displaystyle 2pi} by the absolute value of your coefficient B. The absolute value of B is the non-negative value of B. For example, the absolute value is 3 if B equals 3 or -3. Suppose your sine function is f ( x ) = s i n 2 x {\displaystyle f(x)=sin2x} {\displaystyle f(x)=sin2x}. In this case, B = 2. Plug 2 into the formula to get P e r i o d = 2 p i / | 2 | = 2 p i / 2 = p i {\displaystyle Period=2pi/|2|=2pi/2=pi} {\displaystyle Period=2pi/|2|=2pi/2=pi}. In this case, the period of the sine function is p i {\displaystyle pi} pi. The same steps are used to find the period of a secant function y = A s e c ( B x ) {\displaystyle y=Asec(Bx)} {\displaystyle y=Asec(Bx)} and a cosecant function y = A c s c ( B x ) {\displaystyle y=Acsc(Bx)} {\displaystyle y=Acsc(Bx)}.
If your sine or cosine function is s i n ( x ) {\displaystyle sin(x)} sin(x) or c o s ( x ) {\displaystyle cos(x)} {\displaystyle cos(x)}, the period is always 2 p i {\displaystyle 2pi} {\displaystyle 2pi}. 2 p i {\displaystyle 2pi} {\displaystyle 2pi} is the period for a standard sine or cosine curve. If no other coefficients or variables are introduced to change the period, the period of s i n ( x ) {\displaystyle sin(x)} sin(x) or c o s ( x ) {\displaystyle cos(x)} {\displaystyle cos(x)} will always be 2 p i {\displaystyle 2pi} {\displaystyle 2pi}.
Finding the Period of a Tangent Function
Find coefficient B in tangent function y = A t a n ( B x ) {\displaystyle y=Atan(Bx)} {\displaystyle y=Atan(Bx)}. If your tangent function is y = t a n ( 2 x ) {\displaystyle y=tan(2x)} {\displaystyle y=tan(2x)}, then B = 2. Ignore any other whole numbers in the tangent function; i.e. if the function is y = 8 t a n ( 3 x ) {\displaystyle y=8tan(3x)} {\displaystyle y=8tan(3x)}, you only need to pay attention to 3, which represents coefficient B.
Substitute the value of B into the formula P = p i / | B | {\displaystyle P=pi/|B|} {\displaystyle P=pi/|B|}. Divide p i {\displaystyle pi} pi by the absolute value of B to find the period of the tangent function. For example, suppose your tangent function is y = t a n ( 2 x ) {\displaystyle y=tan(2x)} {\displaystyle y=tan(2x)}. Plug B (2) into the formula to obtain P = p i / | 2 | {\displaystyle P=pi/|2|} {\displaystyle P=pi/|2|}. The absolute value of 2 is 2, which leads us to P = p i / 2 {\displaystyle P=pi/2} {\displaystyle P=pi/2}. Solve the equation to get . The period of this tangent function is . The same steps are used to find the period of a cotangent function y = A c o t ( B x ) {\displaystyle y=Acot(Bx)} {\displaystyle y=Acot(Bx)}.
If the tangent function is t a n ( x ) {\displaystyle tan(x)} {\displaystyle tan(x)}, the period is always p i {\displaystyle pi} pi. A standard tangent function will always have a period of p i {\displaystyle pi} pi. Therefore, you don’t have to calculate the period for function t a n ( x ) {\displaystyle tan(x)} {\displaystyle tan(x)}.
Periodic Function Practice Problems
Example Problem 1: Find the period of the periodic function s i n ( 4 x + 5 ) {\displaystyle sin(4x+5)} {\displaystyle sin(4x+5)}. Solution: B = 4. P e r i o d = 2 p i / | 4 | {\displaystyle Period=2pi/|4|} {\displaystyle Period=2pi/|4|} → 2 p i / 4 {\displaystyle 2pi/4} {\displaystyle 2pi/4} → p i / 2 {\displaystyle pi/2} {\displaystyle pi/2}. The period of s i n ( 4 x + 5 ) {\displaystyle sin(4x+5)} {\displaystyle sin(4x+5)} is p i / 2 {\displaystyle pi/2} {\displaystyle pi/2}.
Example Problem 2: Find the period of the periodic function s i n ( − 8 x ) {\displaystyle sin(-8x)} {\displaystyle sin(-8x)}. Solution: B = – 8. P e r i o d = 2 p i / | − 8 | {\displaystyle Period=2pi/|-8|} {\displaystyle Period=2pi/|-8|} → 2 p i / 8 {\displaystyle 2pi/8} {\displaystyle 2pi/8} → p i / 4 {\displaystyle pi/4} {\displaystyle pi/4}. The period of s i n ( − 8 x ) {\displaystyle sin(-8x)} {\displaystyle sin(-8x)} is p i / 4 {\displaystyle pi/4} {\displaystyle pi/4}.
Example Problem 3: Find the period of the periodic function t a n 3 x {\displaystyle tan3x} {\displaystyle tan3x}. Solution: B = 3. P e r i o d = p i / | 3 | {\displaystyle Period=pi/|3|} {\displaystyle Period=pi/|3|} → p i / 3 {\displaystyle pi/3} {\displaystyle pi/3}. The period of t a n 3 x {\displaystyle tan3x} {\displaystyle tan3x} is p i / 3 {\displaystyle pi/3} {\displaystyle pi/3}.
Example Problem 4: Find the period of the periodic function s e c 2 x + 1 {\displaystyle sec2x+1} {\displaystyle sec2x+1}. Solution: B = 2. P e r i o d = 2 p i / | 2 | {\displaystyle Period=2pi/|2|} {\displaystyle Period=2pi/|2|} → 2 p i / 2 {\displaystyle 2pi/2} {\displaystyle 2pi/2} → p i {\displaystyle pi} pi. The period of s e c 2 x + 1 {\displaystyle sec2x+1} {\displaystyle sec2x+1} is p i {\displaystyle pi} pi.
Example Problem 5: Find the period of the periodic function 5 c o s 12 x {\displaystyle 5cos12x} {\displaystyle 5cos12x}. Solution: B = 12. P e r i o d = 2 p i / | 12 | {\displaystyle Period=2pi/|12|} {\displaystyle Period=2pi/|12|} → 2 p i / 12 {\displaystyle 2pi/12} {\displaystyle 2pi/12} → p i / 6 {\displaystyle pi/6} {\displaystyle pi/6}. The period of 5 c o s 12 x {\displaystyle 5cos12x} {\displaystyle 5cos12x} is p i / 6 {\displaystyle pi/6} {\displaystyle pi/6}.
Example Problem 6: Find the period of the periodic function c o t ( x ) {\displaystyle cot(x)} {\displaystyle cot(x)}. Solution: The period of c o t ( x ) {\displaystyle cot(x)} {\displaystyle cot(x)} is p i {\displaystyle pi} pi because it’s a standard cotangent function.
What is a periodic function?
A periodic function is a function that repeats itself at regular intervals. The period of a function is the distance between each repetition. Periodic functions are represented by the formula f ( x + p ) = f ( x ) {\displaystyle f(x+p)=f(x)} {\displaystyle f(x+p)=f(x)}, where p {\displaystyle p} p is the period of the function and f {\displaystyle f} f is the periodic function. A periodic function is also defined by having a positive real number (i.e., a number greater than zero) to represent p {\displaystyle p} p. Therefore, f ( x + p ) = f ( x ) {\displaystyle f(x+p)=f(x)} {\displaystyle f(x+p)=f(x)} is true for x {\displaystyle x} x being all real numbers. The fundamental period of a function is the lowest possible value of the positive real number p {\displaystyle p} p, or the period in which a function repeats itself. The sine wave is a classic example of a periodic function. The sine wave graph looks like the same wave shape repeated over and over again. The distance between the peaks (or valleys) of each subsequent wave on the graph is the period of the function, as it represents the distance between each repetition.
Periodic functions are also defined by amplitude, phase/vertical shift, and frequency. The common form for graphing periodic functions is f ( x ) = A s i n ( B x + C ) + D {\displaystyle f(x)=Asin(Bx+C)+D} {\displaystyle f(x)=Asin(Bx+C)+D} or f ( x ) = A c o s ( B x + C ) + D {\displaystyle f(x)=Acos(Bx+C)+D} {\displaystyle f(x)=Acos(Bx+C)+D}, where A = amplitude, B = frequency, –C/B = phase shift, and D = vertical shift. While the period of a function defines the distance between each repetition of the curve, these other coefficients define other dimensions of the graph. The amplitude of a periodic function is the height or highest point of each peak in a curve pattern. Amplitude can be found by graphing the function, identifying the minimum and maximum values of the function, then halving that difference. That final number is the amplitude. The frequency of a periodic function is the number of repeated wave patterns within a certain interval. For sine and cosine functions, this interval is often from 0 - 2pi. For tangent functions, it’s typically from 0 - pi. A phase shift is when a curve shifts horizontally on a graph to the left or right of its normal position. The phrase shift does not affect the other variables in a periodic function, i.e. the frequency, period, or amplitude. A vertical shift is when a curve shifts vertically on the graph to be higher or lower than its usual position. If D is negative, the function will shift vertically DOWN the y-axis—if positive, the function will shift upward.
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