Roots of trigonometrical equation

Roots of trigonometric equations are the values of the unknown angle that satisfy the given trigonometric equation. These equations involve trigonometric functions like sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). Understanding how to find and interpret these roots is crucial in solving problems related to angles and periodic functions.

1. Basic Concepts:

  • Definition:

    • A root of a trigonometric equation is an angle for which the equation is satisfied.
  • Periodicity:

    • Trigonometric functions are periodic, repeating their values at regular intervals. Consider the principal period of the function when finding roots.

2. Finding Roots:

  • Isolating the Trigonometric Function:

    • Start by isolating the trigonometric function on one side of the equation.
  • Applying Trigonometric Identities:

    • Use trigonometric identities to simplify the equation or express one trigonometric function in terms of another.
  • Algebraic Techniques:

    • Employ algebraic techniques such as factoring, completing the square, or using quadratic formulas to simplify the equation.
  • Multiple Roots:

    • Consider multiple roots within the given range, especially for periodic functions.

3. Common Trigonometric Equations:

  1. Linear Equations:

    • Example: sin(x)=0
    • Solutions: x=nπ, where n is an integer.
  2. Quadratic Equations:

    • Example: cos2(x)cos(x)=0
    • Solutions: x=0,2π3,π,4π3, and so on.
  3. Inverse Trigonometric Equations:

    • Example: tan1(x)=π4
    • Solutions: x=tan(π4)=1

4. Periodicity and General Solutions:

  • Periodicity of Trigonometric Functions:

    • Sine and cosine have a principal period of 2π, while tangent, cotangent, secant, and cosecant have a principal period of π.
  • General Solutions:

    • Express the roots in terms of general solutions considering the periodic nature of trigonometric functions.