, , , and functions expand on the basic sine and cosine. These trig functions have unique patterns and properties, like periods, ranges, and asymptotes, that set them apart from their simpler cousins.
Understanding how to graph and transform these functions is key. You'll learn about their domains, ranges, and special features like vertical asymptotes. This knowledge helps solve more complex trig problems and model real-world periodic phenomena.
Graphing Tangent, Secant, Cosecant, and Cotangent Functions
Graph of tangent function
Periodic function repeats its pattern every π units
Vertical asymptotes occur at x=2π+nπ where n is any integer, when the cosine function equals 0
Graph passes through the origin (0, 0)
As x approaches vertical asymptotes from the left, tanx approaches positive or negative infinity (depending on the quadrant)
As x approaches vertical asymptotes from the right, tanx approaches negative or positive infinity (opposite of left-hand limit)
Can take on any real value between negative and positive infinity
Tangent is an odd function, exhibiting symmetry about the origin
Transformations of tangent functions
Vertical shifts y=tanx+k move the graph up by k units if k>0, down by ∣k∣ units if k<0
Horizontal shifts y=tan(x−h) move the graph right by h units if h>0, left by ∣h∣ units if h<0
Vertical stretches/compressions y=atanx stretch the graph vertically by a factor of ∣a∣ if ∣a∣>1, compress it if 0<∣a∣<1 (reflection across x-axis if a<0)
Horizontal stretches/compressions y=tan(bx) compress the graph horizontally by a factor of ∣b∣ if ∣b∣>1, stretch it if 0<∣b∣<1 ( becomes ∣b∣π)
The amplitude of is undefined due to its vertical asymptotes
Secant vs cosecant graphs
Secant y=secx has a period of 2π, cosecant y=cscx also has a period of 2π
Both have vertical asymptotes
Secant: at x=2π+nπ where n is any integer (cosine equals 0)
Cosecant: at x=nπ where n is any integer (sine equals 0)
Secant has a minimum of 1 and maximum of -1, cosecant has no max/min values
Transformed secant and cosecant period becomes ∣b∣2π, cotangent period becomes ∣b∣π
Properties of trigonometric functions
Tangent:
Domain: all real numbers except x=2π+nπ where n is any integer
Range: (−∞,∞)
Vertical asymptotes: x=2π+nπ where n is any integer
Secant:
Domain: all real numbers except x=2π+nπ where n is any integer
Range: (−∞,−1]∪[1,∞)
Vertical asymptotes: x=2π+nπ where n is any integer
Cosecant:
Domain: all real numbers except x=nπ where n is any integer
Range: (−∞,−1]∪[1,∞)
Vertical asymptotes: x=nπ where n is any integer
Cotangent:
Domain: all real numbers except x=nπ where n is any integer
Range: (−∞,∞)
Vertical asymptotes: x=nπ where n is any integer
Continuity of trigonometric functions
Tangent, secant, cosecant, and cotangent functions are continuous on their domains
Discontinuities occur at vertical asymptotes where these functions are undefined
Key Terms to Review (10)
Cosecant: The cosecant function, denoted as $\csc(\theta)$, is the reciprocal of the sine function. It is defined as $\csc(\theta) = \frac{1}{\sin(\theta)}$ where $\sin(\theta) \neq 0$.
Cosecant function: The cosecant function is the reciprocal of the sine function. It is defined as $\csc(x) = \frac{1}{\sin(x)}$ for all values of $x$ where $\sin(x) \ne 0$.
Cotangent: Cotangent is a trigonometric function defined as the reciprocal of the tangent function. It can be expressed as $\cot(\theta) = \frac{1}{\tan(\theta)}$ or $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$.
Cotangent function: The cotangent function, denoted as $\cot(x)$, is the reciprocal of the tangent function. It can be defined as $\cot(x) = \frac{1}{\tan(x)}$ or $\cot(x) = \frac{\cos(x)}{\sin(x)}$.
Period: The period of a trigonometric function is the interval over which it completes one full cycle and starts to repeat. For sine and cosine functions, the period is $2\pi$.
Secant: A secant function, denoted as $\sec(\theta)$, is the reciprocal of the cosine function. It is defined as $\sec(\theta) = \frac{1}{\cos(\theta)}$.
Secant function: The secant function, denoted as $\sec(\theta)$, is the reciprocal of the cosine function. It is defined as $\sec(\theta) = \frac{1}{\cos(\theta)}$.
Stretching/compressing factor: A stretching/compressing factor in trigonometric graphs is a coefficient that alters the amplitude or period of the function. It can vertically stretch/compress the graph or horizontally stretch/compress it.
Tangent: A tangent is a trigonometric function represented as $\tan(\theta)$, which is the ratio of the sine and cosine of an angle: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. It is undefined when $\cos(\theta) = 0$.
Tangent function: The tangent function, denoted as $\tan(\theta)$, is a trigonometric function defined as the ratio of the sine and cosine functions: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. It is periodic with a period of $\pi$ radians.