5.3 Solving Trigonometric Equations
Trigonometric equations are solved by standard algebraic techniques (collecting like terms and factoring). The goal is to isolate the trigonometric function involved in the equation.
Solving a Trigonometric Equation
Collecting Like Terms
Extracting Square Roots
If two or more functions occur in the same equation, collect all terms on one side and then attempt to separate the functions by factoring or using appropriate identities.
Factroing
Equations of Quadratic Type
Many trigonometric equations are of quadratic type ax² + bx + c = 0, These equations can be solved by factoring the quadratic, or using the quadratic formula.
When working with an equation of quadratic type, be sure that the equation involves a single trigonometric function.
Sometimes each side of an equation must be squared to obtain a quadratic. Because this procedure can introduce extraneous solutions, the solutions must be checked.
When working with an equation of quadratic type, be sure that the equation involves a single trigonometric function.
Sometimes each side of an equation must be squared to obtain a quadratic. Because this procedure can introduce extraneous solutions, the solutions must be checked.
Functions of Multiple Angles
Trigonometric functions of multiple angles of the forms sin ku and cos ku are solved by solving for ku, and then dividing the result by k.
Trigonometric functions of multiple angles of the forms sin ku and cos ku are solved by solving for ku, and then dividing the result by k.
Using Inverse Functions
With some trigonometric equations, there is no reasonable way to find the solutions algebraically. In those cases, the solutions can still be approximated.
Approximating solutions
Use a graphing utility to graph the function and approximate the maximum and minimum points of the graph in the interval [0, 2π] and solve the trigonometric equation and verify that the x coordinates of the maximum and minimum points of f are among its solutions.
f(x) = sin x + cos x cos x - sin x = 0 |