Chapter 8: Trigonometric Equations and Applications

1. Chapter 8: Trigonometric Equations and Applications L8.5 Solving More Difficult Trig Equations

2. Solving More Difficult Trig Equations A conditional equation is one that is true for only some values of the variable(s). In L8.1, we solved simple eqns w/ only one trig function raised to the 1st power. In this lesson, we consider more complex equations that may have squared functions or more than one type of trig function. The following strategies can be applied to solve. 1. Algebra 1 techniques (collect like terms, etc.) [L8.1] 2. Factoring (do not divide as you may lose a root) 3. Square Root Property (taking the square root of both sides) 4. Converting to one type of trig function by using the Pythagorean Ids 5. Squaring both sides (extraneous solutions possible, so check ans!) 6. Then use the techniques from L8.1 to find the values on the unit circle or use the inverse trig functions. Adjust the interval of interest if necessary and ensure that all solutions are provided. You should have the fundamental identities in front of you when you work!

3. Examples: Solve for the variable w/in [0, 2 π) Factor: 1. cos2θ− 3cosθ = 4 2. sinθ·tanθ = 3 sinθ Sq Root Property: 3. 2sin2x – 1 = 0 Convert to one type of trig fcn, using the fundamental identities: 4. sin2θ – sinθ = cos2θ 5. 2cot2θ + 3cscθ = 1 6. 2sinθ = cosθ Square both sides (always check for extraneous solutions): 7. sec x + 1 = tan x 8. 2sinθ = cosθ + 1 Convert to sine & cosine [#7 can be done this way too] 9. sec x = 2 csc x Recall that solns can be checked by plugging them in!