Read the Six Trigonometric Graphs

Unroll the circle into a sine wave

Let the horizontal axis show an angle $x$ in radians and the vertical axis show the output $\sin x$. These two axes have different jobs: the input is an angle; the output is the vertical coordinate of a point on the unit circle. As the radius turns, record that coordinate at successive angles.

The first five landmarks are

Between them the sine function varies smoothly. From 0 to $\pi/2$, the circle point rises; from $\pi/2$ to $3\pi/2$, its vertical coordinate falls; then it rises again towards 0. The graph records this changing height, not the circular path itself. A sine graph is not drawn by joining a few landmarks with straight lines.

Every real angle has a circle point, so the domain (परिभाषा-क्षेत्र) is $\mathbb R$. Its vertical coordinate can be any number between −1 and 1, including both endpoints, so the range (मान-क्षेत्र) is $[-1,1]$. The graph repeats after $2\pi$; its smallest positive period is $2\pi$.

Zeros occur at $x=n\pi$, where $n$ is any integer. The graph is positive when the terminal point is above the horizontal axis and negative when it is below. Read the sign over an interval, not just at one endpoint. These landmarks let you test a sketch: if its first positive peak is labelled π rather than $\pi/2$, the horizontal scale is wrong.

Cosine starts at the rightmost point

Cosine records the unit-circle point’s horizontal coordinate. At $x=0$, the point is $(1,0)$, so the cosine graph begins at output 1 rather than 0. At a quarter-turn its horizontal coordinate becomes 0; at a half-turn it reaches−1. This produces the landmarks

Like sine, cosine is defined for every real input and has range $[-1,1]$. Its smallest positive period is $2\pi$. The maximum 1 occurs at $x=2n\pi$, and the minimum−1 at $x=(2n+1)\pi$. Zeros lie at $x=\pi/2+n\pi$. Here and throughout these descriptions, $n\in\mathbb Z$.

Reflecting an angle across the horizontal axis leaves the point’s horizontal coordinate unchanged. Thus $\cos(-x)=\cos x$: cosine is even, and its graph is symmetric about the vertical axis. Compare this with sine’s origin symmetry, $\sin(-x)=-\sin x$. Symmetry tells you how to extend a carefully drawn part of a graph without guessing its negative-input behaviour.

Sine and cosine have the same wave shape with different starting positions. The identity $\cos x=\sin(x+\pi/2)$ expresses this as a phase shift. Do not read it as $\cos x=\sin x+\pi/2$; one changes the input angle, while the other would add to the output. You can verify the distinction immediately at $x=0$.

Tangent has branches, not a continuous wave

Tangent is the quotient $\tan x=\sin x/\cos x$. It is defined only when $\cos x\ne0$. Therefore the graph has no points at

As an input approaches one of these excluded positions from a side, the quotient’s magnitude grows without bound. Draw a dashed vertical asymptote there and break the curve. An asymptote is a guide to nearby behaviour, not part of the function graph. A graphing display must never join the upper end of one branch to the lower end of another with a vertical stroke.

On the central interval $(-\pi/2,\pi/2)$, tangent increases through $(-\pi/4,-1)$, $(0,0)$ and $(\pi/4,1)$. It takes every real output on this interval alone. Its range is $\mathbb R$, and it has no maximum or minimum on its full domain. Infinity is not an output value reached at an excluded input.

A half-turn changes both sine and cosine signs, leaving their quotient unchanged. Thus $\tan(x+\pi)=\tan x$, and the smallest positive period is π. Tangent is odd: $\tan(-x)=-\tan x$. Its zeros are $x=n\pi$. These three facts—zeros, excluded inputs and one correctly shaped branch—are enough to build the repeating graph accurately.

When reading a plotted value, first check the input is allowed. Near a pole, rounded decimal values may be very large; exactly at a pole, a display should say undefined instead of showing a large finite approximation.

Cotangent changes which coordinate is underneath

Cotangent is $\cot x=\cos x/\sin x$. The denominator now vanishes at $x=n\pi$, so these inputs are excluded. In particular, cotangent is undefined at 0 even though tangent is 0 there. The two functions are related, but their domains are not interchangeable.

Consider the interval $(0,\pi)$. Cotangent decreases from arbitrarily large positive values near 0 to 0 at $\pi/2$, then to arbitrarily large negative values near π. Useful landmarks are $(\pi/4,1)$, $(\pi/2,0)$ and $(3\pi/4,-1)$. Draw the branch between, rather than on, its two vertical asymptotes.

The full domain is $\mathbb R\setminus\{n\pi:n\in\mathbb Z\}$, its range is $\mathbb R$, and its smallest positive period is π. Its zeros are $\pi/2+n\pi$. It is odd because $\cot(-x)=-\cot x$ wherever both sides are defined. As with tangent, the full-domain function is unbounded and has no attained maximum or minimum.

The reciprocal statement $\cot x=1/\tan x$ is useful only where both expressions exist. At $x=\pi/2$, for example, $\cot x=0$, while $\tan x$ is undefined. You cannot start with the undefined tangent and take its reciprocal to obtain a valid calculation. Use $\cos x/\sin x$, the defining quotient, when checking an axis value.

A strong graph comparison focuses on the denominators: tangent’s poles are cosine’s zeros; cotangent’s poles are sine’s zeros. This also explains why the branches appear shifted rather than merely flipped reciprocals of every plotted point.

Secant and cosecant leave a gap in their ranges

Secant and cosecant are reciprocal functions:

They inherit domain exclusions from the zeros of their denominators. Secant is undefined at $x=\pi/2+n\pi$; cosecant is undefined at $x=n\pi$. Draw vertical asymptotes at those inputs and keep the graph branches separated.

Because sine and cosine lie between −1 and 1, a nonzero denominator with magnitude at most 1 has a reciprocal with magnitude at least 1. Consequently both functions have range

They never produce outputs strictly between −1 and 1, including 0. The endpoints−1 and 1 are included: they occur when the original denominator is −1 or 1. The curved branches touch these boundary levels; they do not cross into the empty band between them.

Secant has a positive branch centred at 0, passing through $(0,1)$, and a negative branch centred at π, passing through $(\pi,-1)$. Cosecant has a positive branch between 0 and π, passing through $(\pi/2,1)$, and a negative branch between π and $2\pi$, passing through $(3\pi/2,-1)$.

Both repeat after $2\pi$, their smallest positive period. Secant is even because cosine is even; cosecant is odd because sine is odd. Neither has a global maximum or minimum on its full domain: each is unbounded above and below. The values 1 and−1 are turning points of individual branches, not the global extremes of the entire function.

To sketch either function, first draw a faint guide for its denominator. Mark that guide’s zeros and extrema, then build the reciprocal branches without crossing the asymptotes.

Read a graph in a dependable order

Before calculating a point, ask what inputs are allowed. Then read the graph’s zeros, sign, range and repetition. This order prevents an attractive sketch from hiding a forbidden denominator. Use the explorer to select a function and move its input marker; compare each readout with an exact angle you already know.

A period is a positive input shift that preserves every output and the pattern of allowed inputs. It must work across the domain, not just at two convenient points. Although sine repeats after $4\pi$, that is not its smallest positive period. Although some sine values repeat after π, the whole function does not: $\sin(\pi/2)=1$ while $\sin(3\pi/2)=-1$.

Check the axes as well. A horizontal tick labelled π means an angle of π radians, not an output height. At an asymptote, the moving marker should disappear or identify the value as undefined. Between asymptotes it should remain on the selected continuous branch. Finally, compare the graph with its symmetry: sine, tangent, cotangent and cosecant are odd; cosine and secant are even. Together these checks form a small but powerful graph-reading routine.

Use symmetry, then explore one small change

A function is even when $f(-x)=f(x)$ for every allowed input and its negative. Its graph reflects in the vertical axis. A function is odd when $f(-x)=-f(x)$; its graph has a half-turn symmetry about the origin. A picture suggests these properties, while the coordinate identities prove them.

For example, secant is even because $\sec(-x)=1/\cos(-x)=1/\cos x=\sec x$. Tangent is odd because $\sin(-x)/\cos(-x)=-\sin x/\cos x$. Their excluded inputs are also symmetric, so the statements apply to their entire domains.

A graph can also help compare outputs on a stated interval. On $0\le x\le\pi$, sine and cosine meet at $\pi/4$. Before that input cosine is higher; afterwards sine is higher. This is an interval observation, not a claim that one function is always larger. Always write the interval beside a graph comparison.

Explore further: changing a wave. Once the six basic graphs are secure, compare $y=\sin x$ with $y=2\sin x$, $y=\sin2x$ and $y=|\sin x|$. The first doubles the output height; the second completes a cycle twice as quickly; the third reflects negative output upward. For $\sin2x$, the range stays $[-1,1]$ while the period becomes π. For $|\sin x|$, the range is $[0,1]$ and the period is also π.

These changes are an optional exploration beyond reading the six basic graphs. Notice the parentheses: $|\sin x|$ and $\sin|x|$ are different functions. Moving a symbol changes the mathematical instruction, so the explorer and the written formula must agree.