Trigonometric Equations

The equations containing trigonometric functions or t-ratios of an unknown angle or real number are known as trigonometric equations.

Example:

cos x = ½, sin x = 0, tan x = √3 etc. are trigonometric equations.

Solution of a trigonometric equation:

A solution of a trigonometric equation is the value of the unknown angle that satisfies the equation.

Example:

Consider the equation sin θ = ½. This equation is, clearly, satisfied by θ = \(\frac{π}{6}\), \(\frac{5π}{6}\) etc. so these its solution. Solving an equation means to find the set of all values of the unknown value which satisfy the given equation.

The solutions lying between 0 to 2π or between 0° to 360° are called principal solutions.

Clearly we see that principal solution of the equation sin θ = ½ are π/6 and 5π/6 because these solutions lie between 0 to 2π.

Consider the equation 2 cos θ + 1 = 0 or cos θ = -1/2. This equation is clearly, satisfy by θ = \(\frac{2π}{3}\), \(\frac{4π}{3}\) etc. Since the trigonometric functions are periodic, therefore, if a trigonometric equation has a solution, it will have infinitely number of solutions. For example, θ = \(\frac{2π}{3}\), 2π ± \(\frac{2π}{3}\), 4π ± \(\frac{2π}{3}\), ………… are solutions of  2 cos θ + 1 = 0. These solutions can be put together in compact form as 2nπ ± \(\frac{2π}{3}\) where n is an integer. This solution is known as the general solution. Thus, a solution generalize by means of periodicity is known as the general solution.

It also follows from the above discussion that solving an equation means to find its general solution.

Let us observe the difference in the relation involving one or more trigonometrical function

(i) sin\(^{2}\) x + cos\(^{2}\) x = 1

This relation is satisfied by all real values of x for which the function sin x and cos x are defined. Such a relation involving one or more trigonometrical function is called trigonometrical identity.

(ii) sin θ = -5

We know the range of sin θ is - 1 ≤ sin θ ≤ 1. Therefore, there is no real value of θ will satisfy the equation sin θ = -5

(iii) 2 sin x = 1

This relation is not satisfied by any value of the angle x; It is satisfied by a definite set of value of x. Such a relation involving one or more trigonometrical function which is satisfied by a definite set of value (finite or infinite) of the associated angle (or angles) is called a trigonometrical equation.






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