**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 the **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 a trigonometric function**, we
take the angle x along the X-axis and the value of trigonometric ratio sinx, cosx,
tanx, etc. along the 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 the 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 the sine function (Sine Graph).

If we study this sine curve, we will get the 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 the 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 the 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 the cosine function (Cosine Graph).

If we study this cosine curve, we will get the 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 the 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 the 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:

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 the cotangent function (Cotangent Graph).

*Do you have any questions regarding the graph of a trigonometric function?*

*Do you have any questions regarding the graph of a trigonometric function?*

*You
can ask your questions or problems here, in the comment section below.*

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