hiw many solution can we find in principal solution and general solution ????
Consider the equation sin θ = ½. This equation is, clearly, satisfied by θ = π6, 5π6 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 = 2π/3, 4π/3etc. Since the trigonometric functions are periodic, therefore, if a trigonometric equation has a solution, it will have infinitely number of solutions. For example, θ = 2π/3, 2π ± 2π/3, 4π ± 2π/3, ………… are solutions of 2 cos θ + 1 = 0. These solutions can be put together in compact form as 2nπ ± 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.
Consider the equation sin θ = ½. This equation is, clearly, satisfied by θ = π6, 5π6 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 = 2π/3, 4π/3etc. Since the trigonometric functions are periodic, therefore, if a trigonometric equation has a solution, it will have infinitely number of solutions. For example, θ = 2π/3, 2π ± 2π/3, 4π ± 2π/3, ………… are solutions of 2 cos θ + 1 = 0. These solutions can be put together in compact form as 2nπ ± 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.
