Lesson 5.3. Solving Trigonometric Equations

Transcription

1 Lesson 5.3 Solving

2 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.

3 For all problems, The solution interval Will be [0, π) You are responsible for checking your solutions back into the original problem!

4 Solve: cos 1= 0 Step 1: Isosolate cos using algebraic skills. cos = 1 cos = 1 Step : 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., 3 3 Note: The reference angle is π/3.

5 Solve: Step 1: Step : Solving tan 1 = 0 tan = 1 tan = ± 1 tan = ± 1 = Q1 QII QIII QIV π 3,,, Note: Since there is a ±, all four quadrants hold a solution with π/4 being the reference angle.

6 Step 1: Solve: cot cos = cot cot cos cot = 0 ( ) cot cos = 0 cot = 0 or cos = 0 cos = cos =± cos =± π 3π =, = Step : Note: There is no solution here because ± lies outside the range for cosine.

7 Try these: tan + 1 = 0 sec 4 = 0 3 3tan = tan Solution 3π =, 4 4 π π 4 =,,, π = 0,,, π,,

8 Solve: sin sin 1= 0 ( )( ) sin + 1 sin 1 = 0 sin+ 1= 0 or sin 1= 0 1 sin = sin = 1 11π π =, = 6 6 Factor the quadratic equation. Set each factor equal to zero. Solve for sin Determine the correct quadrants for the solution(s).

9 Solve: sin + 3cos 3 = 0 ( ) 1 cos + 3cos 3 = 0 cos + 3cos 3= 0 cos + 3cos 1 = 0 cos 3cos + 1 = 0 ( )( ) cos 1 cos 1 = 0 cos 1 = 0 or cos 1 = 0 1 cos = cos = 1 =, = Replace sin with 1-cos Distribute Combine like terms. Multiply through by 1. Factor. Set each factor equal to zero. Solve for cos. Determine the solution(s).

10 Solve: cos + 1 = sin ( cos + 1) = ( sin ) Square both sides of the equation in order to change sine into terms of cosine giving only one trig function to work with. cos + cos + 1 = sin cos + cos + 1 = 1 cos cos + cos = 0 ( ) cos cos+ 1 = 0 cos = 0 or cos + 1 = 0 cos = 0 cos = 1 π 3π, X = = π Why is 3π/ removed as a solution? FOIL or Double Distribute Replace sin with 1 cos Set equation equal to zero since it is a quadratic equation. Factor Set each factor equal to zero. Solve for cos Determine the solution(s). It is removed because it does not check in the original equation.

11 Solution: Solve: cos3 = 1 No algebraic work needs to be done because cosine is already by itself. Remember, 3 refers to an angle and one cannot divide by 3 because it is cos 3 which equals ½. Since 3 refers to an angle, find the angles whose cosine value is ½. 3 =, 3 3 =, π 3 =,,, π =,,, Now divide by 3 because it is angle equaling angle. Notice the solutions do not eceed π. Therefore, more solutions may eist. Return to the step where you have 3 equaling the two angles and find coterminal angles for those two. Divide those two new angles by 3.

12 11π 13π 1 3 =,,,,, The solutions still do not eceed π. Return to 3 and find two more coterminal angles. 11π 13π 1 =,,,,, Divide those two new angles by π 13π 1 19π 3π The solutions still do not eceed π. 3 =,,,,,,, Return to 3 and find two more coterminal angles. 11π 13π 1 19π =,,,,,, Notice that 19π/9 now eceeds π and is not part of the solution. Therefore the solution to cos 3 = ½ is Divide those two new angles by 3. 11π 13π 1 =,,,,,

13 Try these: 4sin = cos + 1 csc + cot = 1 3 sin = Solution = π = 5π 11π =,,, cos = = π

14 What you should know: 1. How to use algebraic techniques to solve trigonometric equations.. How to solve quadratic trigonometric equations by factoring or the quadratic formula. 3. How to solve trigonometric equations involving multiple angles.