Solving Trigonometric Equations

1. Use all algebraic techniques learned in Algebra II. 2. Look for factoring and collecting like terms. 3. Isolate the trig function in the equation. 4. Use the inverse trig functions to assist in determining solutions. 5. Identify the CORRECT (local or global) solution!

Example Solve cos 2x = 1 – sin x for 0 ≤ x < 2π

Solve for 0 o ≤θ<360 o cos 2x = cos x

Example Solve 2sin2x = 1 for 0 o ≤ θ < 360 o Pretend the 2 isn’t in front of the x and solve it (solve sin x = ½ )

Example: Solve 2 cos 2 x  1 = 0 Solution: First, solve for cos x

Example: Solve 2 cos x + sec x = 0 Solution Since neither factor of the equation can equal zero, the equation has no solution.

Example: Solve: Note: There is no solution here because  2 lies outside the range for cosine.

Example: Solve cos x + 1 = sin x [0, 2  ] Check the solutions in the original equation. The only solutions are  /2 and .

Solve the equation

Solving Trigonometric Equations Solve: Replace sin 2 x with 1-cos 2 x Distribute Combine like terms. Multiply through by – 1. Factor. Set each factor equal to zero. Solve for cos x. Determine the solution(s).

Solving Trigonometric Equations Solve: Square both sides of the equation in order to change sine into terms of cosine giving only one trig function to work with. FOIL or Double Distribute Replace sin 2 x with 1 – cos 2 x Set equation equal to zero since it is a quadratic equation. Factor Set each factor equal to zero. Solve for cos x Determine the solution(s). X Why is 3  /2 removed as a solution? It is removed because it does not check in the original equation.

Solving Trigonometric Equations, Finding all Solutions Where n is any integer. An equivalent answer would be

Solve a trigonometric equation

Solve a trigonometric equation by factoring