Solution of trigonometric equations

The General solution of trigonometric equations of the type sin y = sin a, cos y = cos a and tan y = tan a

General Solution: The entire set of the values belonging to the unknown angle satisfies the equation. Also, it has all the specific solutions along with the principal solutions.

General Solutions of a Trigonometric Equation

From the below-given diagram, we can observe that sin(π -θ) = sin θ & cos ( -θ) = cos θ.

We use the above description for finding the solutions of a few trigonometric equations.

Now, Solve sin(x) = y for x.

Case 1: When the condition is -1≤y≤ 1, i.e, y’s value is between

(-1 & 1), so, there’s the solution.

sin(x) = y is the set of all the possible solutions which is:

x = sin^-1(y) + 2kπ & x = −sin^-1(y) + (2k+1) π, Where “k” can be taken as any integer; i.e, solutions for x is having sin^-1(y) in addition to all the even multiples of the value π, together with the minus sin^-1(y) & all the odd multiples regarding π.

Case 2: When -1 > y or y > 1, i.e, y’s value is very much large or very much small for the possible solution.

There’re no solutions.

Solving Cos (x) = y for x.

Case 1: -1≤y≤ 1

cos(x) = y is the set of all the solutions which is equal to

x = ± cos^-1(y) + 2kπ,

Where “k” can be taken as any integer;

Case 2: When, -1 > y or y > 1

There exist no solutions.

Solving tan(x) = y for x.

Tan(x) = y equation is the set of all solution to:

x = tan^-1(y) + kπ,

Where the value of “k” can be taken as “any integer”

Example

Find the general solutions of √ 2 sinx = 1.

The equation is equivalent to Sinx = $\frac{1}{√2}$

Now, Sin ⁻¹ = $\frac{\pi }{4}$

Hence the general solution is

x = ±π/4 + 2kπ, where the value of k can be any integer.

Trigonometric Equations with their General solutions