VIEWS: 12 PAGES: 14 POSTED ON: 8/9/2012 Public Domain
Lesson 5.3 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 Solve: 2cos x 1 0 Step 1: Isosolate cos x using algebraic skills. 2cos x 1 cos x 1 2 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. 5 Note: The reference angle is /3. x , 3 3 Solving Trigonometric Equations Solve: tan x 1 0 2 Step 1: tan x 1 2 tan x 1 2 Note: Since there is a , all four quadrants tan x 1 hold a solution with /4 being the reference angle. Step 2: QIV Q1 QII QIII 3 5 7 x , , , 4 4 4 4 Solving Trigonometric Equations Solve: cot x cos x 2cot x 2 Step 1: cot x cos2 x 2cot x 0 cot x cos2 x 2 0 cot x 0 or cos 2 x 2 0 cos2 x 2 cos2 x 2 cos x 2 3 x Note: There is no solution here because 2 Step 2: x , lies outside the range for cosine. 2 2 Solving Trigonometric Equations Try these: Solution 3 7 1. tan x 1 0 x , 4 4 2 4 5 2. sec x 4 0 2 x 3 , 3 , 3 , 3 5 7 11 3. 3tan x tan x 3 x 0, 6 , 6 , , 6 , 6 Solving Trigonometric Equations Solve: 2sin 2 x sin x 1 0 2sin x 1sin x 1 0 Factor the quadratic equation. 2sin x 1 0 or sin x 1 0 Set each factor equal to zero. 1 sin x sin x 1 Solve for sin x 2 7 11 x , x Determine the correct quadrants 6 6 2 for the solution(s). Solving Trigonometric Equations Solve: 2sin 2 x 3cos x 3 0 2 1 cos2 x 3cos x 3 0 Replace sin2x with 1-cos2x 2 2cos2 x 3cos x 3 0 Distribute 2cos2 x 3cos x 1 0 Combine like terms. 2cos2 x 3cos x 1 0 Multiply through by – 1. 2cos x 1 cos x 1 0 Factor. 2cos x 1 0 or cos x 1 0 Set each factor equal to zero. 1 cos x cos x 1 Solve for cos x. 2 5 x , x0 Determine the solution(s). 3 3 Solving Trigonometric Equations Solve: cos x 1 sin x Square both sides of the equation cos x 1 sin x 2 2 in order to change sine into terms of cosine giving only one trig function to work with. cos2 x 2cos x 1 sin 2 x FOIL or Double Distribute cos2 x 2cos x 1 1 cos2 x Replace sin2x with 1 – cos2x 2cos2 x 2cos x 0 Set equation equal to zero since it is a quadratic equation. 2cos x cos x 1 0 Factor 2cos x 0 or cos x 1 0 Set each factor equal to zero. cos x 0 cos x 1 Solve for cos x 3 x ,X x Determine the solution(s). 2 2 Why is 3/2 removed as a solution? It is removed because it does not check in the original equation. Solving Trigonometric Equations 1 Solve: cos 3x 2 No algebraic work needs to be done because cosine is already by itself. Solution: Remember, 3x refers to an angle and one cannot divide by 3 because it is cos 3x which equals ½. Since 3x refers to an angle, find the angles whose cosine value is ½. 5 3x , Now divide by 3 because it is angle equaling angle. 3 3 5 Notice the solutions do not exceed 2. Therefore, x , 9 9 more solutions may exist. 5 7 11 Return to the step where you have 3x equaling 3x , , , the two angles and find coterminal angles for 3 33 3 those two. 5 7 11 x , , , Divide those two new angles by 3. 9 9 9 9 Solving Trigonometric Equations 5 7 11 13 17 The solutions still do not exceed 2. 3x , , , , , Return to 3x and find two more 3 3 3 3 3 3 coterminal angles. 5 7 11 13 17 x , , , , , Divide those two new angles by 3. 9 9 9 9 9 9 5 7 11 13 17 19 23 The solutions still do not exceed 2. 3x , , , , , , , Return to 3x and find two more 3 3 3 3 3 3 3 3 coterminal angles. 5 7 11 13 17 19 x , , , , , , Divide those two new angles by 3. 9 9 9 9 9 9 9 Notice that 19/9 now exceeds 2 and is not part of the solution. 5 7 11 13 17 Therefore the solution to cos 3x = ½ is x , , , , , 9 9 9 9 9 9 Solving Trigonometric Equations Try these: Solution 1. 4sin 2 x 2cos x 1 x 5.4218 2. csc x cot x 1 x 2 3 2 5 5 11 3. sin 2 x x , , , 2 3 6 3 6 x 2 4. cos x 2 2 2 Solving Trigonometric Equations What you should know: 1. How to use algebraic techniques to solve trigonometric equations. 2. How to solve quadratic trigonometric equations by factoring or the quadratic formula. 3. How to solve trigonometric equations involving multiple angles.