**Graphs
of Trigonometric Functions**

To each
angle x there corresponds a unique value for the trigonometric ratios and hence
the trigonometric ratios such as sinx, cosx, tanx including the reciprocal
ratios are **functions** of the angle x.
A function can be represented in various ways. One of the important ways by
which a function can be represented is the **graph**.
In a **trigonometric function**, the
variable is the angle. For different values of the angle, the values of the
trigonometric ratios will be different. To draw the **graph of trigonometric function**, we take the angle x along X-axis
and the value of trigonometric ratio sinx, cosx, tanx, etc. along Y-axis.
Taking different values of the angle as the x-coordinates, and the
corresponding values of the trigonometric ratio as y-coordinates, we plot the
points (x, sinx), etc. on the plane with rectangular axes. Then we join these
points freely to get required graph of the trigonometric function.

Consider a
circle with radius r placed in standard position. Let a revolving line OP start
from OX and trace out an angle XOP = x°. Draw perpendicular PM from P to the
X-axis.

__1 ^{st} quadrant__:OP revolves
from OX to OY, x varies from 0° to 90°, MP varies from 0 to r, MP is positive,
OM varies from r to 0 and OM is positive.

__2 ^{nd} quadrant__:OP revolves
from OY to OX’, x varies from 90° to 180°, MP varies from r to 0, MP is
positive, OM varies from 0 to – r and OM is negative.

__3 ^{rd} quadrant__:OP revolves
from OX’ to OY’, x varies from 180° to 270°, MP varies from 0 to – r, MP is
negative, OM varies from – r to 0 and OM is negative.

__4 ^{th} quadrant__:OP revolves
from OY’ to OX, x varies from 270° to 360°, MP varies from – r to 0, MP is
negative, OM varies from 0 to r, OM is positive.

**Graph
of Sine Function (sine graph)**

The sine function is defined by y =
sinx = MP/OP = MP/r.

When x
varies from 0° to 90°, y = sinx varies from 0 to 1 because MP varies from 0 to
r. When x varies from 90° to 180°, y = sinx varies from 1 to 0 because MP
varies from r to 0. When x varies from 180° to 270°, y = sinx varies from 0 to
– 1 because MP varies from 0 to – r. When x varies from 270° to 360°, y = sinx
varies from – 1 to 0.

Some
standard values of x and the corresponding values of sinx are given in the
table below:

Plotting these values in the graph paper, we get the following graph of sine function (sine graph).

If we study this sine curve, we will
get following facts:

(i)
y
= sinx attains the maximum value at 90° and minimum value at 270°. The maximum
and minimum value of sinx are 1 and – 1 respectively. So y = sinx oscillates in
the limit – 1 to 1.

(ii)
y
= sinx has positive values in the first and second quadrants where the graph is
above X-axis and negative values in the third and fourth quadrants where the
graph is below X-axis.

**Graph
of Cosecant Function (cosecant graph)**

Similarly,
we can have the table of trigonometric values and graph of the reciprocal of
sine function i.e cosecant function (y = cosecx) as given below:

Plotting these values in the graph paper, we get the following graph of cosecant function (cosecant graph).

**Graph
of Cosine Function (cosine graph)**

The cosine
function is defined by y = cosx = OM/OP = OM/r.

When x
increases from 0° to 90°, cosx decreases from 1 to 0 as OM decreases from r to
0. When x increases from 90° to 180°, cosx decreases from 0 to – 1 because OM
changes 0 to – r. When x changes from
180° to 270°, cosx changes from – 1 to 0. Similarly, when x increases from 270°
to 360°, cosx increases from 0 to 1.

Some
standard values of x and the corresponding values of cosx are given in the
table below:

Plotting these values in the graph paper, we get the following graph of cosine function (cosine graph).

If we study this cosine curve, we
will get following facts:

(i)
y
= cosx attains the maximum value at x = 0° and 360° and minimum value at 180°.
The maximum and minimum value of cosx are 1 and – 1 respectively. So y = cosx oscillates
in the limit – 1 to 1.

(ii)
y
= cosx has positive values in the first and fourth quadrants where the graph is
above X-axis and negative values in the second and third quadrants where the
graph is below X-axis.

**Graph
of Secant Function (secant graph)**

Similarly,
we can have the table of trigonometric values and graph of the reciprocal of cosine
function i.e secant function (y = secx) as given below:

Plotting these values in the graph paper, we get the following graph of secant function (secant graph).

**Graph
of Tangent Function (tangent graph)**

The tangent
function is defined by y = tanx = MP/OM

When x
increases from 0° to 90°, tanx increases from 0 to ∞ (infinity) because MP
changes from 0 to r and OM changes from r to 0. When x increases from 90° to
180°, tanx changes from - ∞ to 0 because MP changes from r to 0 and OM changes
from 0 to – r. When x increases from 180° to 270°, tanx changes from 0 to ∞
because MP changes from 0 to – r and OM changes from – r to 0. When x increases
from 270° to 360°, tanx changes from - ∞ to 0 because MP changes from – r to 0 and
OM changes from 0 to r.

Some standard values of x and the corresponding values of tanx are given in the table below:

Plotting these values in the graph paper, we get the following graph of tangent function (tangent graph).

y = tanx has positive values in the first and third quadrants where the graph is above X-axis and negative values in the second and fourth quadrants where the graph is below X-axis. The maximum and minimum values of tanx cannot be defined.

**Graph
of Cotangent Function (cotangent graph)**

Similarly,
we can have the table of trigonometric values and graph of the reciprocal of tangent
function i.e cotangent function (y = cotx) as given below:

Plotting these values in the graph paper, we get the following graph of cotangent function (cotangent graph).

*You can
comment your questions or problems regarding the graphs of trigonometric
functions.*

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