Solving Trigonometric Equations For all problems The solution

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 Solve: Step 1: Isolate cos x. 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). I IV 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: Q 1 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: 1. 2. 3. Solution

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).

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 X Why is 3 /2 removed as a solution? Determine the solution(s). It is removed because it does not check in the original equation.

Solving Trigonometric Equations Solve: Solution: No algebraic work needs to be done because cosine is already by itself. Remember, 3 x refers to an angle and one cannot divide by 3 because it is cos 3 x which equals ½. Since 3 x 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 3 x 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 3 x 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 3 x = ½ is

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