Trigonometric Functions-I, Graphs of Trigonometric Functions

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Graphs of Trigonometric Functions

  • Given any function, a pictorial or a graphical representation makes a lasting impression on the minds of learners and viewers.

  • The importance of the graph of functions stems from the fact that this is a convenient way of presenting many properties of the functions.

  • By observing the graph we can examine several characteristic properties of the functions such as (i) periodicity, (ii) intervals in which the function is increasing or decreasing (iii) symmetry about axes, (iv) maximum and minimum points of the graph in the given interval.

  • It also helps to compute the areas enclosed by the curves of the graph.

Variations of Sin Θ as Θ Varies Continuously from 0 to 2π

  • Let and be the axes of coordinates. With centre and radius

  • unity, draw a circle. Let starting from OX and moving in anticlockwise direction make an angle with the -axis, i.e.. Draw , then as .

  • The variations of are the same as those of .

I Quadrant:

I Quadrant of sinθ

I Quadrant of Sinθ

I Quadrant of sinθ

  • As increases continuously from to .

  • is positive and increases from to .

  • is positive.

II Quadrant

II Quadrant of sinθ

II Quadrant of Sinθ

II Quadrant of sinθ

  • In this interval, lies in the second quadrant.

  • Therefore, point is in the second quadrant. Here

  • is positive, but decreases from to as varies from to .

  • is positive.

III Quadrant

III Quadrant of sinθ

III Quadrant of Sinθ

III Quadrant of sinθ

  • In this interval, lies in the third quadrant.

  • Therefore, point can move in the third quadrant only.

  • Hence is negative and decreases from to as varies from to .

  • In this interval decreases from to .

  • is negative.

IV Quadrant

IV Quadrant of sinθ

IV Quadrant of Sinθ

IV Quadrant of sinθ

  • In this interval, lies in the fourth quadrant.

  • Therefore, point can move in the fourth quadrant only.

  • Here again is negative but increases from to as varies from to .

  • is negative.

Graph of as varies from to .

  • Let and be the two coordinate axes of reference. The values of are to be measured along -axis and the values of are to be measured along -axis.

  • (Approximate value of )

Graph of sinθ as θ varies from 0 to 2π

Graph of Sinθ as Θ Varies from 0 to 2π

Graph of sinθ as θ varies from 0 to 2π

Example: Draw the graph of in the interval to .

Solution:

The graph of y=sin2θ

The Graph of Y=sin2θ

The graph of y=sin2θ

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