1. Solving Trigonometric Equations MATH 109 - Precalculus S. Rook

2. Overview Section 5.3 in the textbook: – Basics of solving trigonometric equations – Solving linear trigonometric equations – Solving quadratic trigonometric equations – Solving trigonometric equations with multiple angles – Solving other types of trigonometric equations – Approximate solutions to trigonometric equations 2

3. Basics of Solving Trigonometric Equations

4. To solve a trigonometric equation when the trigonometric function has been isolated: – e.g. – Look for solutions in the interval 0 ≤ θ < period using the unit circle Recall the period is 2π for sine, cosine, secant, & cosecant and π for tangent & cotangent We have seen how to do this when we discussed the circular trigonometric functions in section 4.2 – If looking for ALL solutions, add period ∙ n to each individual solution Recall the concept of coterminal angles 4

5. Basics of Solving Trigonometric Equations (Continued) – We can use a graphing calculator to help check (NOT solve for) the solutions E.g. For, enter Y1 = sin x, Y2 =, and look for the intersection using 2 nd → Calc → Intersect 5

6. Basics of Solving Trigonometric Equations (Example) Ex 1: Find all solutions and then check using a graphing calculator: 6

7. Solving Linear Trigonometric Equations

8. Solving Linear Equations Recall how to solve linear algebraic equations: – Apply the Addition Property of Equality Isolate the variable on one side of the equation Add to both sides the opposites of terms not associated with the variable – Apply the Multiplication Property of Equality Divide both sides by the constant multiplying the variable (multiply by the reciprocal) 8

9. Solving Linear Trigonometric Equations An example of a linear equation: Solving trigonometric linear (first degree) equations is very similar EXCEPT we: – Isolate a trigonometric function of an angle instead of a variable Can view the trigonometric function as a variable by making a substitution such as Revert to the trigonometric function after isolating the variable – Use the Unit Circle and/or reference angles to solve 9

10. Solving Linear Trigonometric Equations (Example) Ex 2: Find all solutions: 10

11. Solving Quadratic Trigonometric Equations

12. Recall a Quadratic Equation (second degree) has the format – One side MUST be set to zero Common methods used to solve a quadratic equation: – Factoring Remember that the process of factoring converts a sum of terms into a product of terms – Usually into two binomials – Quadratic Formula 12

13. Solving Quadratic Trigonometric Equations (Continued) The same methods can be used to solve a quadratic trigonometric equation: – Substituting a variable for a trigonometric function is acceptable so long as there is only one trigonometric function present in the equation e.g. Let y = tan x – Be aware of extraneous solutions if fractions are present Those solutions which cause the denominator to equal 0 13

14. Solving Quadratic Trigonometric Equations (Example) Ex 3: Solve in the interval 0 ≤ x < 2π: a) b) c) 14

15. Trigonometric Equations with Two Different Trigonometric Functions Be aware when a quadratic trigonometric equation exists with two DIFFERENT trigonometric functions – Not like Example 3c because after factoring out tan x, the equation became two linear trigonometric equations – Recall how we handled two different trigonometric functions in section 5.1 15

16. Trigonometric Equations with Two Different Trigonometric Functions (Continued) If we have two different trigonometric functions raised to the first power: – Square both sides and apply Pythagorean identities to simplify the equation E.g. – Recall that when we square both sides of an equation some of the potential solutions will not check into the original equation MUST check all solutions into the original problem Discard those solutions that do not check 16

17. Trigonometric Equations with Two Different Trigonometric Functions (Example) Ex 4: Solve in the interval 0 ≤ x < 2π: a) b) c) 17

18. Solving Trigonometric Equations with Multiple Angles

19. A trigonometric equation with a multiple angle has the form kx where k ≠ 1 (a single- angle trigonometric function otherwise) To solve a trigonometric equation with multiple-angles e.g. 1 + cos 3x = 3 ⁄ 2 : – Isolate the trigonometric function either by solving for it or applying a quadratic strategy: e.g. cos 3x = ½ 19

20. Solving Trigonometric Equations with Multiple Angles (Continued) – Find all solutions in the interval of [0, period) e.g. – Isolate the variable: e.g. – If necessary, let n vary to find all solutions in the interval [0, 2π): e.g. 20

21. Solving Trigonometric Equations with Multiple Angles (Example) Ex 5: Find all solutions in the interval [0, 2π): 21

22. Other Types of Trigonometric Equations

23. Trigonometric Equations and the Sum & Difference Formulas Recall the sum and difference formulas – Valid in both directions Given a trigonometric equation involving the right-hand side of a sum or difference formula: – Condense into the left-hand side of the formula e.g. – Use previously discussed strategies to solve 23

24. Trigonometric Equations and Multiple-Angle Formulas Recall the double-angle and half-angle formulas – We can use either the left or right sides of these formulas Overall goal is to isolate the trigonometric function 24

25. Other Types of Trigonometric Equations (Example) Ex 6: Solve in the interval [0, 2π): a) b) sin 6x + sin 2x = 0 c) 4 sin x cos x = 1 d) 25

26. Approximate Solutions to Trigonometric Equations

27. More often than not we run into solutions of trigonometric equations that are NOT one of the special values on the unit circle Solve as normal until the trigonometric function is isolated Calculate the reference angle Use the reference angle AND the sign of the value of the trigonometric function to estimate the solutions in the interval [0, period) 27

28. Approximate Solutions to Trigonometric Equations (Example) Ex 7: Find all solutions in the interval [0, 2π) – use a calculator to estimate: a) b) 28

29. Summary After studying these slides, you should be able to: – Solve linear trigonometric equations – Solve quadratic trigonometric equations – Solve trigonometric equations with multiple angles – Solve other types of trigonometric equations including sum & difference formulas, double-angle & half-angle formulas – Approximate the solutions to trigonometric equations Additional Practice – See the list of suggested problems for 5.3 Next lesson – Law of Sines (Section 6.1) 29