The trigonometric ratios to any angle in terms of radian measure are called trigonometric functions.

We have studied the principles of trigonometric ratios in the lower classes using acute angles . But we come across many angles which are not acute. We shall extend the acute angle idea and define trigonometric functions for any angle. The trigonometric ratios to any angle in terms of radian measure are called trigonometric functions.

Let P(x, y) be a point other than the origin on the terminal side of an angle θ in standard position . Let OP = r.

Thus, r = √(x2 + y2)

The six trigonometric functions of θ are defined as follows:

Consider the unit circle x2 +y2 = 1. Let P(x, y) be a point on the unit circle where the terminal side of the angle θ intersects the unit circle.

Thus, the coordinates of any point P(x, y) on the unit circle is (cos θ,sin θ). In this way, the angle measure θ is associated with a point on the unit circle.

The following table illustrates how trigonometric function values are determined for a Quadrantal angles using the above explanation.

Consider a unit circle with the centre at the origin. Let the angle zero (in radian measure) be associated with the point

For each real number

Now, define sin

Clearly, sin

Using sin

Consider a unit circle with centre at the origin. Let θ be in standard position. Let P(x, y) be the point on the unit circle corresponding to the angle θ. Then, cos θ = x, sin θ = y and tan θ = y/x.

The values of x and y are positive or negative depending on the quadrant in which P lies.

cos θ = x > 0 (positive); sin θ = y > 0 (positive)

Thus, cos θ and sin θ and hence all trigonometric functions are positive in the first quadrant.

cos θ = x < 0 (negative); sin θ = y > 0 (positive)

Thus, sin θ and cosec θ are positive and others are negative.

Similarly, we can find the sign of trigonometric functions in other two quadrants.

Let us illustrate the above discussions in Figure 3.10.

Signs of trigonometric functions in various quadrants can be remembered with the slogan

“All Students Take Chocolate”. (ASTC rule)

We know that a function

For example, sin (

Similarly, cos

But tan

The periodicity of sin

Here

Basic trigonometric functions are examples of non-polynomial even and odd functions

Because cos(

Also note that sec

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11th Mathematics : UNIT 3 : Trigonometry : Trigonometric functions and their properties | Mathematics

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