The Central Idea
A trigonometric function assigns a numerical output to an angle.
For example,
\[ y=\sin x \]assigns to each allowable input \(x\) its sine value.
On the unit circle,
\[ \sin x \]is the vertical coordinate,
\[ \cos x \]is the horizontal coordinate, and
\[ \tan x=\frac{\sin x}{\cos x}. \]A graph displays these outputs continuously as the angle changes.
From Circular Motion to a Graph
Imagine a point rotating around the unit circle.
At each angle \(x\), record the point's vertical coordinate.
Plot that vertical coordinate against the angle:
\[ (x,\sin x). \]As the rotating point rises, falls, crosses the horizontal axis, and repeats its motion, the plotted points form the sine wave.
Recording the horizontal coordinate instead produces the cosine wave.
Angle Input on the Horizontal Axis
In a trigonometric graph, the horizontal coordinate usually represents angle measure rather than ordinary geometric distance.
We may write inputs in degrees,
\[ 0^\circ,\quad90^\circ,\quad180^\circ,\quad270^\circ,\quad360^\circ, \]or in radians,
\[ 0,\quad\frac{\pi}{2},\quad\pi,\quad\frac{3\pi}{2},\quad2\pi. \]Radian measure is standard for higher mathematics, but the geometric behavior is the same.
Periodicity
A function is periodic if its values repeat after a fixed horizontal interval.
Sine and cosine satisfy
\[ \sin(x+2\pi)=\sin x \]and
\[ \cos(x+2\pi)=\cos x. \]Their basic period is
\[ \boxed{2\pi} \]radians, equivalent to
\[ \boxed{360^\circ}. \]The Sine Function
The basic sine function is
\[ \boxed{y=\sin x}. \]On the unit circle, \(\sin x\) is the \(y\)-coordinate.
As the angle moves through one revolution, sine follows the sequence
\[ 0,\quad1,\quad0,\quad-1,\quad0. \]These values occur at quarter-turn intervals.
Key Sine Points over One Cycle
\[ \begin{array}{c|ccccc} x & 0 & \frac{\pi}{2} & \pi & \frac{3\pi}{2} & 2\pi\\ \hline \sin x & 0 & 1 & 0 & -1 & 0 \end{array} \]Therefore the sine graph passes through
\[ (0,0), \quad \left(\frac{\pi}{2},1\right), \quad (\pi,0), \quad \left(\frac{3\pi}{2},-1\right), \quad (2\pi,0). \]Shape of the Sine Graph
Beginning at the origin, the sine graph:
- rises smoothly to \(1\);
- falls smoothly back to \(0\);
- continues downward to \(-1\);
- rises back to \(0\);
- repeats the same pattern.
The curve is smooth rather than made of line segments.
Domain of Sine
The unit circle defines sine for every real angle.
Therefore,
\[ \boxed{\operatorname{domain}(\sin)=(-\infty,\infty)}. \]Range of Sine
The vertical coordinate of a point on the unit circle can never exceed \(1\) or fall below \(-1\).
Therefore,
\[ \boxed{-1\leq\sin x\leq1}. \]The range is
\[ \boxed{[-1,1]}. \]Zeros of Sine
Sine is zero whenever the unit-circle point lies on the \(x\)-axis.
This occurs at integer multiples of \(\pi\):
\[ \boxed{x=k\pi,\qquad k\in\mathbb Z}. \]These inputs are the \(x\)-intercepts of the sine graph.
Maximum and Minimum of Sine
The maximum value of sine is
\[ \boxed{1}, \]occurring at
\[ x=\frac{\pi}{2}+2\pi k. \]The minimum value is
\[ \boxed{-1}, \]occurring at
\[ x=\frac{3\pi}{2}+2\pi k. \]Sine Is an Odd Function
The unit circle gives
\[ \sin(-x)=-\sin x. \]Therefore the sine function is odd.
Its graph has rotational symmetry about the origin:
\[ \boxed{\sin(-x)=-\sin x}. \]The Cosine Function
The basic cosine function is
\[ \boxed{y=\cos x}. \]On the unit circle, cosine is the horizontal coordinate.
During one revolution, cosine follows
\[ 1,\quad0,\quad-1,\quad0,\quad1. \]Key Cosine Points over One Cycle
\[ \begin{array}{c|ccccc} x & 0 & \frac{\pi}{2} & \pi & \frac{3\pi}{2} & 2\pi\\ \hline \cos x & 1 & 0 & -1 & 0 & 1 \end{array} \]Therefore cosine passes through
\[ (0,1), \quad \left(\frac{\pi}{2},0\right), \quad (\pi,-1), \quad \left(\frac{3\pi}{2},0\right), \quad (2\pi,1). \]Shape of the Cosine Graph
Beginning at \(x=0\), the cosine graph:
- starts at its maximum value \(1\);
- falls smoothly through \(0\);
- reaches \(-1\);
- rises through \(0\);
- returns to \(1\);
- repeats.
Domain and Range of Cosine
Cosine is defined for every real angle:
\[ \boxed{\operatorname{domain}(\cos)=(-\infty,\infty)}. \]Its output is a unit-circle \(x\)-coordinate, so
\[ \boxed{\operatorname{range}(\cos)=[-1,1]}. \]Zeros of Cosine
Cosine is zero where the unit-circle point lies on the \(y\)-axis.
Thus,
\[ \boxed{ x=\frac{\pi}{2}+k\pi, \qquad k\in\mathbb Z }. \]Maximum and Minimum of Cosine
Cosine reaches its maximum value
\[ 1 \]at
\[ \boxed{x=2\pi k}. \]It reaches its minimum value
\[ -1 \]at
\[ \boxed{x=\pi+2\pi k}. \]Cosine Is an Even Function
The unit circle gives
\[ \cos(-x)=\cos x. \]Therefore cosine is even.
Its graph is symmetric about the \(y\)-axis:
\[ \boxed{\cos(-x)=\cos x}. \]Sine and Cosine Have the Same Shape
Sine and cosine trace the same basic wave pattern, but they begin at different locations in the cycle.
In fact,
\[ \boxed{ \sin x=\cos\left(x-\frac{\pi}{2}\right) } \]and
\[ \boxed{ \cos x=\sin\left(x+\frac{\pi}{2}\right) }. \]A later lesson will describe such horizontal shifts systematically as phase shifts.
Comparing Sine and Cosine at \(x=0\)
At
\[ x=0, \]sine begins at
\[ \sin0=0, \]while cosine begins at
\[ \cos0=1. \]This difference reflects the unit-circle point
\[ (1,0). \]The Tangent Function
The basic tangent function is
\[ \boxed{y=\tan x}. \]Since
\[ \tan x=\frac{\sin x}{\cos x}, \]tangent depends on the ratio of the sine and cosine graphs.
Where Tangent Is Undefined
Tangent is undefined whenever
\[ \cos x=0. \]Therefore,
\[ \boxed{ x=\frac{\pi}{2}+k\pi } \]are excluded from the domain.
These excluded inputs become vertical asymptotes of the tangent graph.
Vertical Asymptotes of Tangent
During a basic interval around the origin, tangent has vertical asymptotes at
\[ x=-\frac{\pi}{2} \]and
\[ x=\frac{\pi}{2}. \]Between them, the tangent graph increases continuously from very large negative values to very large positive values.
Key Tangent Points
Three useful points in one central tangent branch are
\[ \left(-\frac{\pi}{4},-1\right), \qquad (0,0), \qquad \left(\frac{\pi}{4},1\right). \]The branch lies between the asymptotes
\[ x=-\frac{\pi}{2} \qquad\text{and}\qquad x=\frac{\pi}{2}. \]Period of Tangent
A half-turn changes
\[ (\cos x,\sin x) \]into
\[ (-\cos x,-\sin x). \]Their ratio remains unchanged:
\[ \frac{-\sin x}{-\cos x} = \frac{\sin x}{\cos x}. \]Therefore,
\[ \boxed{\tan(x+\pi)=\tan x}. \]The basic period of tangent is
\[ \boxed{\pi}. \]Domain of Tangent
Tangent is defined for all real numbers except where cosine is zero.
\[ \boxed{ \operatorname{domain}(\tan) = \left\{ x\in\mathbb R: x\neq\frac{\pi}{2}+k\pi \right\} }. \]Range of Tangent
Between consecutive vertical asymptotes, tangent passes through every real output value.
Therefore,
\[ \boxed{\operatorname{range}(\tan)=(-\infty,\infty)}. \]Zeros of Tangent
Tangent is zero when
\[ \sin x=0 \]while cosine is nonzero.
Thus,
\[ \boxed{x=k\pi}. \]Tangent Is an Odd Function
Since
\[ \tan(-x)=-\tan x, \]tangent is odd.
Its graph has rotational symmetry about the origin.
Tangent Does Not Have a Maximum or Minimum
Unlike sine and cosine, tangent is not bounded.
Near each vertical asymptote, tangent can become arbitrarily large in magnitude.
Therefore the basic tangent function has no global maximum or minimum.
What Is a Vertical Asymptote?
A vertical asymptote is a vertical line that a graph approaches while the function values grow without bound in magnitude.
For tangent,
\[ x=\frac{\pi}{2}+k\pi \]marks inputs where the function itself is undefined.
The graph approaches these lines but does not contain points on them.
Why Tangent Has Asymptotes
Near an angle where
\[ \cos x=0, \]the denominator in
\[ \tan x=\frac{\sin x}{\cos x} \]becomes very small.
Dividing by a very small nonzero number produces a value with large magnitude.
The asymptote is therefore a direct consequence of the quotient definition.
The Cosecant Function
Cosecant is the reciprocal of sine:
\[ \boxed{y=\csc x=\frac1{\sin x}}. \]Its graph is therefore controlled by the sine graph.
Domain of Cosecant
Cosecant is undefined wherever sine is zero.
Since sine vanishes at
\[ x=k\pi, \]the domain is
\[ \boxed{ \operatorname{domain}(\csc) = \{x\in\mathbb R:x\neq k\pi\} }. \]Vertical Asymptotes of Cosecant
Every zero of sine creates a vertical asymptote of cosecant.
Therefore,
\[ \boxed{x=k\pi} \]are vertical asymptotes of the cosecant graph.
Range of Cosecant
Since
\[ -1\leq\sin x\leq1, \]the reciprocal cannot lie strictly between \(-1\) and \(1\).
Therefore,
\[ \boxed{ \operatorname{range}(\csc) = (-\infty,-1]\cup[1,\infty) }. \]How Cosecant Relates to Sine
Where
\[ \sin x=1, \]we have
\[ \csc x=1. \]Where
\[ \sin x=-1, \]we have
\[ \csc x=-1. \]Between sine zeros, the cosecant branches curve away from the interval \((-1,1)\).
The Secant Function
Secant is the reciprocal of cosine:
\[ \boxed{y=\sec x=\frac1{\cos x}}. \]Its graph is organized around the cosine graph in the same way that cosecant is organized around sine.
Domain of Secant
Secant is undefined wherever cosine is zero:
\[ \boxed{ x=\frac{\pi}{2}+k\pi }. \]Therefore,
\[ \boxed{ \operatorname{domain}(\sec) = \left\{ x\in\mathbb R: x\neq\frac{\pi}{2}+k\pi \right\} }. \]Range of Secant
Because cosine lies between \(-1\) and \(1\), its reciprocal satisfies
\[ \boxed{ \operatorname{range}(\sec) = (-\infty,-1]\cup[1,\infty) }. \]Vertical Asymptotes of Secant
The zeros of cosine become vertical asymptotes of secant:
\[ \boxed{ x=\frac{\pi}{2}+k\pi }. \]How Secant Relates to Cosine
At cosine maxima,
\[ \cos x=1, \]so
\[ \sec x=1. \]At cosine minima,
\[ \cos x=-1, \]so
\[ \sec x=-1. \]Between cosine zeros, secant forms upward- or downward-opening reciprocal branches.
The Cotangent Function
Cotangent may be written
\[ \boxed{ y=\cot x=\frac{\cos x}{\sin x} }. \]It is also the reciprocal of tangent wherever both functions are defined.
Domain of Cotangent
Cotangent is undefined where
\[ \sin x=0. \]Therefore,
\[ \boxed{ \operatorname{domain}(\cot) = \{x\in\mathbb R:x\neq k\pi\} }. \]Vertical Asymptotes of Cotangent
The vertical asymptotes occur at
\[ \boxed{x=k\pi}. \]Zeros of Cotangent
Cotangent is zero when cosine is zero and sine is nonzero.
Therefore,
\[ \boxed{ x=\frac{\pi}{2}+k\pi }. \]Period and Range of Cotangent
Cotangent repeats after a half-turn:
\[ \boxed{\cot(x+\pi)=\cot x}. \]Its basic period is
\[ \boxed{\pi}. \]Its range is all real numbers:
\[ \boxed{(-\infty,\infty)}. \]Shape of a Cotangent Branch
Between two consecutive asymptotes, such as
\[ x=0 \qquad\text{and}\qquad x=\pi, \]the basic cotangent graph decreases continuously from very large positive values to very large negative values.
It crosses the horizontal axis at
\[ x=\frac{\pi}{2}. \]Periods of the Six Functions
\[ \begin{array}{c|c} \text{Function} & \text{Basic period}\\ \hline \sin x & 2\pi\\ \cos x & 2\pi\\ \tan x & \pi\\ \csc x & 2\pi\\ \sec x & 2\pi\\ \cot x & \pi \end{array} \]Reciprocal functions inherit the repeating structure of their partner functions.
Ranges of the Six Functions
\[ \begin{array}{c|c} \text{Function} & \text{Range}\\ \hline \sin x & [-1,1]\\ \cos x & [-1,1]\\ \tan x & (-\infty,\infty)\\ \csc x & (-\infty,-1]\cup[1,\infty)\\ \sec x & (-\infty,-1]\cup[1,\infty)\\ \cot x & (-\infty,\infty) \end{array} \]Bounded and Unbounded Trigonometric Functions
Sine and cosine are bounded because their values remain between \(-1\) and \(1\).
Tangent and cotangent are unbounded because their quotient denominators can approach zero.
Secant and cosecant are also unbounded because they are reciprocals of quantities that can approach zero.
Continuous versus Interrupted Graphs
Sine and cosine are continuous across all real inputs.
Tangent, cotangent, secant, and cosecant are interrupted at inputs where their defining denominators are zero.
Those interruptions appear as vertical asymptotes.
Zeros and Asymptotes Are Connected
Reciprocal and quotient relationships create a useful structural rule:
- zeros of sine become asymptotes of cosecant and cotangent;
- zeros of cosine become asymptotes of secant and tangent.
This makes it possible to sketch several trigonometric graphs from a small number of underlying relationships.
Sine and Cosecant as a Pair
Cosecant is
\[ \frac1{\sin x}. \]Therefore:
- every sine zero becomes a cosecant asymptote;
- every sine maximum at \(1\) becomes a cosecant point at \(1\);
- every sine minimum at \(-1\) becomes a cosecant point at \(-1\).
Cosine and Secant as a Pair
Secant is
\[ \frac1{\cos x}. \]Therefore:
- every cosine zero becomes a secant asymptote;
- every cosine maximum at \(1\) becomes a secant point at \(1\);
- every cosine minimum at \(-1\) becomes a secant point at \(-1\).
Tangent and Cotangent as Reciprocal Partners
Where both are defined,
\[ \tan x\cot x=1. \]Tangent and cotangent therefore exchange large and small magnitudes.
Their zeros and asymptotes are offset from one another:
\[ \tan x=0 \]at \(x=k\pi\), exactly where cotangent is undefined.
Reading Function Values from a Graph
To evaluate a function graphically, locate the input on the horizontal axis, move vertically to the graph, and read the corresponding output.
For example, the sine graph shows
\[ \sin\left(\frac{\pi}{2}\right)=1. \]Exact special-angle values are best obtained analytically, but graphs are useful for estimation and behavior.
Reading Zeros from a Graph
A zero occurs where the graph crosses or touches the horizontal axis.
For sine,
\[ x=k\pi. \]For cosine,
\[ x=\frac{\pi}{2}+k\pi. \]For tangent,
\[ x=k\pi. \]Reading Intervals of Positive and Negative Values
A graph lies above the horizontal axis where the function is positive and below it where the function is negative.
For example, on
\[ 0On
\[ \piUnit-Circle Signs Become Graph Signs
The quadrant sign rules are visible in the graphs.
During one cycle:
- sine is positive through Quadrants I and II;
- sine is negative through Quadrants III and IV;
- cosine is positive through Quadrants I and IV;
- cosine is negative through Quadrants II and III.
The graph is therefore another representation of the same unit-circle structure.
Symmetry Helps with Graphing
Even and odd identities reduce the amount of graphing work.
Cosine is even:
\[ \cos(-x)=\cos x. \]Sine and tangent are odd:
\[ \sin(-x)=-\sin x, \] \[ \tan(-x)=-\tan x. \]These equations predict graph symmetry before individual points are plotted.
Graphing Sine from Key Points
A reliable basic sketch of
\[ y=\sin x \]over one cycle uses the five quarter-cycle points:
\[ (0,0), \quad \left(\frac{\pi}{2},1\right), \quad (\pi,0), \quad \left(\frac{3\pi}{2},-1\right), \quad (2\pi,0). \]Connect them with a smooth wave and repeat periodically.
Graphing Cosine from Key Points
For
\[ y=\cos x, \]use
\[ (0,1), \quad \left(\frac{\pi}{2},0\right), \quad (\pi,-1), \quad \left(\frac{3\pi}{2},0\right), \quad (2\pi,1). \]Again, connect smoothly and repeat.
Graphing Tangent from Asymptotes and Points
To sketch one central branch of
\[ y=\tan x, \]first draw vertical asymptotes at
\[ x=-\frac{\pi}{2} \qquad\text{and}\qquad x=\frac{\pi}{2}. \]Then plot
\[ \left(-\frac{\pi}{4},-1\right), \quad (0,0), \quad \left(\frac{\pi}{4},1\right). \]Draw an increasing curve between the asymptotes and repeat every \(\pi\).
Graphing Cosecant from Sine
To sketch
\[ y=\csc x, \]- sketch or mentally locate the sine graph;
- place asymptotes at every sine zero;
- mark points where sine equals \(1\) or \(-1\);
- draw reciprocal branches outside the strip \(-1
Graphing Secant from Cosine
To sketch
\[ y=\sec x, \]- locate the cosine graph;
- place asymptotes at every cosine zero;
- mark reciprocal anchor points at \(y=1\) and \(y=-1\);
- draw the secant branches away from the interval \((-1,1)\).
Graphing Cotangent
For one basic branch of
\[ y=\cot x, \]place asymptotes at
\[ x=0 \qquad\text{and}\qquad x=\pi. \]Plot
\[ \left(\frac{\pi}{4},1\right), \qquad \left(\frac{\pi}{2},0\right), \qquad \left(\frac{3\pi}{4},-1\right). \]Draw a decreasing branch and repeat every \(\pi\).
Graphs Are Not Drawn to Scale Automatically
A hand sketch should preserve the essential mathematical structure even if it is not perfectly proportional.
Important features include:
- correct intercepts;
- correct maxima and minima;
- correct vertical asymptotes;
- correct period;
- correct symmetry;
- correct increasing or decreasing behavior.
Common Error: Connecting across an Asymptote
If a function is undefined at a vertical asymptote, the graph must not be drawn through that line.
For tangent,
\[ x=\frac{\pi}{2} \]separates two different branches.
Connecting them would falsely imply that the function has a value there.
Common Error: Giving Tangent the Sine Period
Sine repeats every
\[ 2\pi, \]but tangent repeats every
\[ \boxed{\pi}. \]The quotient
\[ \frac{\sin x}{\cos x} \]is unchanged by a half-turn because both numerator and denominator change sign.
Common Error: Assuming Sine and Cosine Are Unbounded
Sine and cosine are unit-circle coordinates.
Therefore,
\[ -1\leq\sin x\leq1 \]and
\[ -1\leq\cos x\leq1. \]Their basic graphs never exceed those vertical bounds.
Common Error: Giving Secant the Same Range as Cosine
Cosine has range
\[ [-1,1]. \]Secant is its reciprocal, so its range is
\[ \boxed{ (-\infty,-1]\cup[1,\infty) }. \]Secant never takes values strictly between \(-1\) and \(1\).
Common Error: Confusing Zeros with Undefined Inputs
For tangent,
\[ x=k\pi \]gives zeros.
But
\[ x=\frac{\pi}{2}+k\pi \]gives undefined inputs and vertical asymptotes.
A zero is a valid output equal to \(0\); an undefined input is not part of the graph.
Common Error: Plotting Trigonometric Inputs as Ordinary Numbers without Units
The input axis represents angle.
If radians are being used,
\[ \frac{\pi}{2} \]means one quarter-turn, not approximately \(1.57\) units of unrelated linear distance.
Keeping the geometric meaning of the input helps preserve the connection to the unit circle.
Common Error: Treating Reciprocal Graphs as Reflections
The graph of
\[ y=\frac1{\sin x} \]is not obtained by simply reflecting the sine graph.
Taking a reciprocal changes output values according to
\[ y\mapsto\frac1y. \]Zeros become undefined, values near zero become large in magnitude, and \(\pm1\) remain fixed.
A Reliable Procedure for Sketching Sine or Cosine
- Identify one full period.
- Mark the five quarter-cycle inputs.
- Plot the corresponding values \(0,\pm1\).
- Connect the points smoothly.
- Extend the pattern periodically in both directions.
- Check symmetry, zeros, maxima, minima, and range.
A Reliable Procedure for Sketching Tangent or Cotangent
- Identify two consecutive vertical asymptotes.
- Mark the zero midway between the appropriate asymptotes.
- Plot useful special-angle points such as values at \(\pi/4\).
- Determine whether the branch increases or decreases.
- Draw the branch without crossing its asymptotes.
- Repeat the pattern every \(\pi\).
A Reliable Procedure for Sketching Secant or Cosecant
- Identify the partner graph: cosine for secant, sine for cosecant.
- Locate the partner's zeros.
- Draw vertical asymptotes at those inputs.
- Locate where the partner equals \(1\) or \(-1\).
- Use those points as reciprocal anchors.
- Draw branches outside the interval \((-1,1)\).
- Repeat according to the \(2\pi\) period.
Practice 1: Sine Key Values
Give the values of
\[ \sin0,\qquad \sin\frac{\pi}{2},\qquad \sin\pi,\qquad \sin\frac{3\pi}{2},\qquad \sin2\pi. \]Practice 2: Cosine Key Values
Give the values of
\[ \cos0,\qquad \cos\frac{\pi}{2},\qquad \cos\pi,\qquad \cos\frac{3\pi}{2},\qquad \cos2\pi. \]Practice 3: Sine Domain and Range
State the domain and range of
\[ y=\sin x. \]Practice 4: Cosine Zeros
Write the general form of all zeros of
\[ y=\cos x. \]Practice 5: Tangent Asymptotes
Write the general form of the vertical asymptotes of
\[ y=\tan x. \]Practice 6: Tangent Period
What is the basic period of
\[ y=\tan x? \]Practice 7: Cosecant Range
State the range of
\[ y=\csc x. \]Practice 8: Secant Asymptotes
At which inputs is
\[ y=\sec x \]undefined?
Practice 9: Cotangent Zero
Give the general form of the zeros of
\[ y=\cot x. \]Practice 10: Identify the Function
A periodic function has:
- period \(\pi\);
- zeros at \(x=k\pi\);
- vertical asymptotes at \(x=\frac{\pi}{2}+k\pi\);
- range \((-\infty,\infty)\).
Which basic trigonometric function is it?
Practice 11: Reciprocal Structure
Explain why every zero of sine becomes a vertical asymptote of cosecant.
Practice 12: Identify the Error
A student sketches
\[ y=\tan x \]as one continuous curve crossing through
\[ x=\frac{\pi}{2}. \]Explain the error.
Practice Answers
Practice 1
\[ \boxed{ 0,\quad1,\quad0,\quad-1,\quad0 }. \]Practice 2
\[ \boxed{ 1,\quad0,\quad-1,\quad0,\quad1 }. \]Practice 3
Domain:
\[ \boxed{(-\infty,\infty)}. \]Range:
\[ \boxed{[-1,1]}. \]Practice 4
\[ \boxed{ x=\frac{\pi}{2}+k\pi, \qquad k\in\mathbb Z }. \]Practice 5
\[ \boxed{ x=\frac{\pi}{2}+k\pi, \qquad k\in\mathbb Z }. \]Practice 6
\[ \boxed{\pi}. \]Practice 7
\[ \boxed{ (-\infty,-1]\cup[1,\infty) }. \]Practice 8
Secant is undefined where cosine is zero:
\[ \boxed{ x=\frac{\pi}{2}+k\pi, \qquad k\in\mathbb Z }. \]Practice 9
\[ \boxed{ x=\frac{\pi}{2}+k\pi, \qquad k\in\mathbb Z }. \]Practice 10
These properties describe
\[ \boxed{y=\tan x}. \]Practice 11
Cosecant is
\[ \csc x=\frac1{\sin x}. \]At every sine zero, the denominator becomes zero. Division by zero is undefined, so those inputs become vertical asymptotes of the cosecant graph.
Practice 12
At
\[ x=\frac{\pi}{2}, \]cosine is zero, so
\[ \tan x=\frac{\sin x}{\cos x} \]is undefined.
Therefore,
\[ \boxed{x=\frac{\pi}{2}} \]is a vertical asymptote, and the graph must not cross or contain a point there.
What to Remember
The unit circle generates the basic trigonometric graphs.
- Sine and cosine are smooth, bounded waves with period \(2\pi\).
- Sine and cosine both have range \([-1,1]\).
- Tangent has period \(\pi\), range all real numbers, and vertical asymptotes where cosine is zero.
- Cotangent has period \(\pi\), range all real numbers, and vertical asymptotes where sine is zero.
- Cosecant is the reciprocal of sine and has range \((-\infty,-1]\cup[1,\infty)\).
- Secant is the reciprocal of cosine and has the same reciprocal range.
- Zeros of sine create asymptotes for cosecant and cotangent.
- Zeros of cosine create asymptotes for secant and tangent.
- Cosine is even; sine and tangent are odd.
- Graph structure reflects the same rotation, coordinates, signs, and periodicity already seen on the unit circle.
Where We Go Next
The six basic graphs establish the standard shapes and repeating patterns of trigonometric functions.
The next lesson, Amplitude, Period, Frequency, and Phase, studies how these graphs change when they are stretched, compressed, shifted, reflected, and used to model repeating phenomena.