Solving Trigonometric Equations

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

First Degree Trigonometric Equations: These are equations where there is one kind of trig function in the equation and that function is raised to the first power.

Now figure out where sin = -1/2 on the unit circle.

Complete the List of Solutions: If you are not restricted to a specific interval and are asked to give the general solutions then remember that adding on any integer multiple of 2π represents a co-terminal angle with the equivalent trigonometric ratio.

Where k is an integer and gives all the coterminal angles of the solution.

Practice Solve the equation. Find the general solutions

Solving Trigonometric Equations Solve: Step 1: Isosolate cos x using algebraic skills. Step 2: Determine in which quadrants cosine is positive. Use the inverse function to assist by finding the angle in Quad I first. Then use that angle as the reference angle for the other quadrant(s). QI QIV Note: cosine is positive in Quad I and Quad IV. Note: The reference angle is /3.

Solving Trigonometric Equations Solve: Step 1: Note: Since there is a  , all four quadrants hold a solution with /4 being the reference angle. Step 2: Q1 QII QIII QIV

Solving Trigonometric Equations 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 1. 2. 3.

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

This is a difference of squares and can factor Solve each factor and you should end up with 4 solutions

Practice Find the general solutions for

Writing in terms of 1 trig fnc If there is more than one trig function involved in the problem, then use your identities. Replace one of the trig functions with an identity so there is only one trig function being used

Solve the following Replace cos2 with 1 - sin2

Solving Trigonometric Equations 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 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 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. Return to the step where you have 3x equaling the two angles and find coterminal angles for those two. Divide those two new angles by 3.

Solving Trigonometric Equations The solutions still do not exceed 2. Return to 3x and find two more coterminal angles. Divide those two new angles by 3. The solutions still do not exceed 2. Return to 3x and find two more coterminal angles. Divide those two new angles by 3. Notice that 19/9 now exceeds 2 and is not part of the solution. Therefore the solution to cos 3x = ½ is

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

Find the general solutions for sin 3x +2= 1

Practice