Solving Trig Equations

Solving Trigonometric Equations

Homogenous Trigonometric Equations

To solve equations with a single trigonometric function, begin by isolating the function if necessary.
Draw a circle with labelled quadrants and use it to determine all potential solutions.

Non-Homogenous Trigonometric Equations

These are equations with more than one different trigonometric function. Convert these to homogenous equations using key identities such as tan(x) = sin(x)/cos(x) or the Pythagorean identities.

Trigonometric Equations with Coefficients

If the equation includes a coefficient on the angle, for example sin(2x), this means the function completes 2 cycles in the span typically covered by one cycle. Adjust your solutions accordingly.

Trigonometric Equations with Phase Shift

A phase shift is indicated by an addition or subtraction inside the function, e.g., sin(x + π/3). Adjust your solutions to account for these shifts.

Systems of Trigonometric Equations

If presented with more than one equation, solve them as a system, much like linear or quadratic systems.
Use substitution or elimination methods to find solutions that satisfy all equations in the system.

Quadratic Trigonometric Equations

Apply your knowledge from solving quadratic equations. Factor, complete the square, or use the quadratic formula.
Remember that the solutions to the equation sin^2(x) = a are the solutions to sin(x) = √a and sin(x) = -√a.

Remember: Always express your solution in the interval requested in the question, often between 0 and 2π unless specify otherwise. Ensure you find all possible solutions within the given interval.