Trigonometric Equations

Basics of Trigonometric Equations: Trigonometric equations are mathematical equations that involve trigonometric functions of unknown angles.
Methods to solve simple trigonometric equations include using basic identities, using known values of the unit circle, or using inverse trigonometric functions to solve for unknown angles.
Manipulation of Trigonometric Equations: Often, you may have to manipulate the original equation into a new form before you can apply solving methods. This could be conducting simple rearrangement, or using trigonometric identities like the Pythagorean identity, double-angle formula, or half-angle formula to simplify the equation.
Solving the Equation: Once the equation has been manipulated, it can be solved like any other algebraic equation. Remember that solutions to trigonometric equations often involve finding multiple possible angles as solutions.

General Solution of Trigonometric Equations

General Solution Concept: In trigonometry, the concept of a general solution is used to express all solutions of a trigonometric equation. Since trigonometric functions are periodic, they repeat their values in regular intervals or periods. Hence, an equation may have infinite solutions.
The general solution always includes the principal solution (solutions within one period of the function) plus n times the period of the function, where n can be any integer.
The principal solution is the set of solutions that fall within the defined range of the inverse trigonometric function used. The general solution includes the principal solutions plus all other solutions resulting from the periodicity of the trigonometric function.
For sine and cosine equations, as their functions repeat every 360° or 2π radians, the principle solutions are found between 0 and 360° or 0 and 2π for radian measures. For the equation involving tangent function, as it repeats every 180° or π radians, the principal solutions are found between 0 and 180° or between 0 and π for radian measures.

Working with Degrees and Radians

Degrees and Radians: These are the two units of measuring angles that you’ll need to be familiar with when dealing with trigonometric equations. Degree measure is based on the circle being 360°, while radian measure is based on the circle being 2π.
Conversion between degrees and radians is critical when solving trigonometric equations.
The conversion factor between degrees and radians is π/180 or approximately 0.0174533 radians per degree. To convert an angle from degrees to radians, multiply the degree measure by π/180. To convert an angle from radians to degrees, multiply the radian measure by 180/π.

Remember, while solving trigonometric equations, always check your answers back into the initial equation to verify you have correct solutions. This may also help catch extraneous solutions which are mathematically correct but fall outside the domain of the original problem.