General Solution Trigonometric Functions

A solution of a trigonometrical equation is the value of the unknown angle that satisfies the equation. A trigonometrical equation may have infinite number of solutions. Or in other words the equations containing functions of unknown angles are known as trigonometric equations. The solution in which the absolute value of the angle is the least is called principal solution. The expression involving integer n which gives all solutions of a trigonometrical equation is called the general solution.

When a trigonometrical equation is solved, among all solutions the solution which is in [-π /2,π /2] for sine, in [[-π /2,π /2] for tangent and in[0,π ] for cosine are the principal values of those functions. Solving an equation means to find the set of all values of the unknown angle which satisfies the given equation.

Since the trigonometric functions are periodic, if a trigonometric equation has a solution, it will have infinitely many solutions.

This solution is known as the general solution. Thus a solution generalized by means of periodicity is known as the general solution. Hence solving an equation means to find its general solution.

General solution of sine , cos and tan values if π = 0 is

θ = n π, n ∈ z, θ = (2 ⁿ⁺¹) π /2,n ∈ z and θ = n π ,n ∈ z

In order to find the general solutions of trigonometrical equations of the form sin θ= sinα , cos θ= cosα , and tan θ= tanα we can follow the following steps:

• Find a value θ, preferably between 0 and 2 π or between -π and π, satisfying the given equation and call it α
•  General solution of sin values if sin θ= sinα is np+ (-1) n.α, where n ∈ Z

 General solution of sin values if is 2n+α, n ∈ Z General solution of tan values if tan θ = tan α is np + α, n ∈ Z Suppose if the equation is of the form acos θ+bsin θ = c, where a,b,c ∈ R.such that |c|≤√(a²+b²) To solve this type of equations, we reduce them in the form of sin θ= sinα or cos θ= cosα . Then we can apply the general equation rule.