Solving Trigonometric Equations Section 5.5 Solving Trigonometric Equations

Solving Trigonometric Equations To solve trigonometric equations: Use standard algebraic techniques learned in Algebra II. Look for factoring and collecting like terms. Isolate the trig function in the equation. Use the inverse trig functions to assist in determining solutions.

Solving Trigonometric Equations For all problems, The solution interval Will be [0, 2) You are responsible for checking your solutions back into the original problem!

Solving Trigonometric Equations 1. Solve: Step 1: Isolate cos x using algebraic skills. Step 2: Solve using the correct quadrants on the unit circle. QI QIV

Solving Trigonometric Equations Try this: # 22 Solution:

Solving Trigonometric Equations 2. Solve: Step 1: Step 2: Q1 QII QIII QIV

Solving Trigonometric Equations Try this: # 48 Solution:

Solving Trigonometric Equations 3. Solve: Step 1: Step 2: Note: There is no solution here because 2 lies outside the range for cosine.

Solving Trigonometric Equations Try these: Solution #56 #70

Solving Trigonometric Equations 4. Solve: Factor the quadratic equation. Set each factor equal to zero. Solve for sin x Determine the correct quadrants for the solution(s).

Solving Trigonometric Equations 5. Solve: Replace sin2x with 1-cos2x 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 6. 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 sin2x with 1 – cos2x Set equation equal to zero since it is a quadratic equation. Factor Set each factor equal to zero. Solve for cos x X Determine the solution(s). It is removed because it does not check in the original equation. Why is 3/2 removed as a solution?

Solving Trigonometric Equations 7. Solve: No algebraic work needs to be done because cosine is already by itself. Remember, 3x refers to an angle and one cannot divide by 3 because it is cos 3x which equals ½. Solution: Since 3x refers to an angle, find the angles whose cosine value is ½. Now divide by 3 because it is angle equaling angle. Notice the solutions do not exceed 2. Therefore, more solutions may exist.

Solving Trigonometric Equations 7. Solve: Solution continues: We could use periodic property of cosine, and write the general form of the solution of this equation using a period for this function: And the general solution for this equation will be: Now we can assign and calculate x – values making sure we do not exceed 2:

Solving Trigonometric Equations Try these: Solution 1) 2) 3) 4)

Solving Trigonometric Equations What you should know: How to use algebraic techniques to solve trigonometric equations. How to solve quadratic trigonometric equations by factoring or the quadratic formula. How to solve trigonometric equations involving multiple angles.