About "Principal value of trigonometric functions"

Principal value of trigonometric functions :

Before going to learn how to find the principal value of trigonometric functions, we need to know about what is principal value.

Principal solution :

The solution in which the absolute value of the angle is the least is called principal solution.

For example the value of cos o° is 1, the value of cos 2π, 4π ,...... are also 1. But the 0 is known as principal value.

Let us see some examples based on the above concept.

Principal value of trigonometric functions - Examples

Example 1 :

Find the principal value of the following

cos x = √3/2

Solution :

cos x = √3/2 > 0

Principal value of x must be in [0, π]. Since cos x is positive the principal value is in the first quadrant.

We have to think about the angle of cos for which we get the value √3/2.

cos π/6 = √3/2 and π/6 ∈ [0, π]

Hence the principal value of x is π/6.

Example 2 :

Find the principal value of the following

cos θ = -√3/2

Solution :

cos θ = -√3/2 < 0

Whenever we have cos θ the principal value of θ must be in [0, π]. We have to choose one of the angles from the first or second quadrant.

Since the value of cos θ is negative, we have to choose the angle from the second quadrant. For that we have to think about the angle of cos for which we get the value √3/2.

cos (π - (π/6)) = cos 5π/6

cos 5π/6 = -√3/2

Hence the principal value of θ is 5π/6.

Example 3 :

Find the principal value of the following

cosec θ = -2/√3

Solution :

cosec θ = -2/√3

sin θ = -√3/2 < 0

θ lies in the third or fourth quadrant. But principal value must be in [-π/2, π/2]

In the first quadrant we get only we get positive values for all trigonometric ratios.So we have to choose one of the angles from 0 to -π/2 that is negative angle.

Now we have to think about the angle of sin for which we get the value √3/2.

sin (-π/3) = -√3/2

Hence the principal value of θ is -π/3.

After having gone through the stuff given above, we hope that the students would have understood "Principal value of trigonometric functions".

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