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How to Find Principal Solution of Trigonometric Functions

Principal Value Of Trigonometric Functions

Principal values of trigonometric functions are solutions of trigonometric functions, for which the θ value lies between 0 < θ < 2π. The value of the trigonometric function repeat after an interval of 2π, and the values lesser than 2π are the principal values of trigonometric functions.

Let us learn more about the method to find the principal value of the trigonometric function, with examples, solutions.

1. What Are The Principal Values Of Trigonometric Functions?
2. How To Find Principal Values Of Trigonometric Functions?
3. General Solution Vs Principal Values Of Trigonometric Functions
4. Examples On Principal Values Of Trigonometric Functions
5. Practice Questions
6. FAQs on Principal Values Of Trigonometric Functions

What Are The Principal Value Of Trigonometric Functions?

The solutions of trigonometric functions, for which the θ value lies between 0 < θ < 2π, are called the principal values of the trigonometric functions. The value of the trigonometric function repeat after an interval of 2π. The following are the standard values of the trigonometric functions.

θ 0° 30° 45° 60° 90°
Sinθ 0 \(\frac{1}{2}\) \(\frac{1}{\sqrt 2}\) \(\frac{\sqrt 3}{2}\) 1
Cosθ 1 \(\frac{\sqrt 3}{2}\) \(\frac{1}{\sqrt 2}\) \(\frac{1}{2}\) 0
Tanθ 0 \(\frac{1}{\sqrt 3}\) 1 \(\sqrt 3\) Undefined
Cotθ Undefined \(\sqrt 3\) 1 \(\frac{1}{\sqrt 3}\) 0
Secθ 1 \(\frac{2}{\sqrt 3}\) \(\sqrt 2\) 2 Undefined
Cosecθ Undefined 2 \(\sqrt 2\) \(\frac{2}{\sqrt 3}\) 1

The higher values of the angles, and lesser than 2π, can be obtained from the above standard values. Here the standard θ values lesser than π/2 are called the acute angles, and all the principal values of trigonometric functions are calculated using these standard values.

How To Find Principal Values Of Trigonometric Functions?

The principal values of trigonometric functions can be computed through a sequence of steps. First, we find the quadrant in which this principle value exists. Here θ is the acute angle and the principal values in the four quadrants can be obtained as follows.

  • In the first quadrant, the angle is either θ, or π/2 - θ.
  • In the second quadrant, the angle is π/2 + θ, or π - θ.
  • In the third quadrant, the angle is π + θ, or 3π/2 - θ.
  • In the fourth quadrant, the angle is 3π/2 + θ, or 2π - θ.

Further for the trigonometric ratio, we can calculate the principal values based on the following rules.

  • Sinθ, Cosecθ is positive in the first and the second quadrant.
  • Tanθ, Cotθ uis positive in the first quadrant and the third quadrant.
  • Cosθ, Secθ is positive in the first quadrant and the fourth quadrant.
  • Here for π - θ, π + θ,, and 2π - θ. the trigonometric ratio remains the same.
  • For π/2 - θ, π/2 + θ, 3π/2 - θ, and 3π/2 + θ the trigonometric ratio changes: Sine changes to Cosine, Tangent changes to Cotangent, and Secant changes to Cosecant.

General Solution Vs Principal Values Of Trigonometric Functions

The general solutions represent all the higher values of θ which can be obtained using the principal values of trigonometric functions. The values of trigonometric functions repeat after an interval of 2π, and all these higher values of trigonometric functions which are of the same value are represented as the general solution of trigonometric functions. The general solutions of a few of the trigonometric functions are as follows. Here θ is the principal value.

  • The general solution of Sinθ is GS = nπ+(-1)nθ
  • The general solution of Cosθ is GS = 2nπ + θ
  • The general solution of Tanθ is GS = nπ + θ

Related Topics on Principal Value Of Trigonometric Functions

The following related topic links would help in an easier understanding of types of vectors.

  • Domain and Range Of Trigonometric Functions
  • Trigonometric Equationsons
  • Inverse Trigonometric Ratios
  • Trigonometric Functions
  • Trigonometric Chart

Solved Examples on Principal Value Of Trigonometric Functions

  1. Example 1: Find the principal value of the trigonometric function of Cosx = \(\dfrac{\sqrt 3}{2}\).

    Solution:

    The given function is Cosx = \(\dfrac{\sqrt 3}{2}\).

    The Cosecant of x is positive in the first quadrant and the fourth quadrant.

    Cosx = Cos\(\dfrac{π}{6}\)

    Cosx= Cos\(2π - \dfrac{π}{6}\)

    Cosx = Cos \(\dfrac{11π}{6}\)

    Therefore, the principal values of Cosx is \(\dfrac{π}{6}\), \(\dfrac{11π}{6}\).

  2. Example 2: Find the principal value of the trigonometric function of Tanx = \(\dfrac{1}{\sqrt 3}\)

    Solution:

    The given trigonometric function is Tanx = \(\dfrac{1}{\sqrt 3}\).

    Tanx = Tan\(\dfrac{π}{6}\)

    The trigonometric function of Tanx is positive in the first quadrant and the third quadrant.

    Tanx = Tan\(π + \dfrac{π}{6}\)

    Tanx = Tan\(\dfrac{7π}{6}\)

    Therefore, the trigonometric function of Tanx has principal values of \(\dfrac{π}{6}\), and \(\dfrac{7π}{6}\).

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Practice Questions on Principal Value Of Trigonometric Function

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FAQs on Principal Value Of Trigonometric Function

What Are The Principal Value Of Trigonometric Function?

The solutions of trigonometric functions, for which the θ value lies between 0 < θ < 2π, are called the principal values of the trigonometric functions. The value of the trigonometric function repeat after an interval of 2π. The principal value of trigonometric function cn be easily computed from the general solution of the trigonometric function.

How To Find The Principal Value Of Trigonometric Functions?

The principal value of the trigonometric function can be calculated by first breaking the angle into a sum of a larger angle and the acute angle. Further after finding the value of the acute angle for the trigonometric function, we can find the principal solution of the trigonometric function.

What Is The Difference Between Principal Value And General Solution Of Trigonometric Functions?

The principal value of the trigonometric function is a value lesser than 2π, and the trigonometric function values repeat after an interval of 2π. All the higher angle values can be represented as a single equation, which is the general solution of the trigonometric function.

How to Find Principal Solution of Trigonometric Functions

Source: https://www.cuemath.com/trigonometry/principal-value-of-trigonometric-functions/