1. Solving Trigonometric Equations Trigonometry MATH 103 S. Rook

2. Overview Section 6.1 in the textbook: – Solving linear trigonometric equations – Solving quadratic 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 i) θ, 0° ≤ θ < 360° ii) all degree solutions 10

11. Solving Linear Trigonometric Equations (Example) Ex 3: Find i) t, 0 ≤ t < 2π ii) all radian solutions 11

12. Solving Linear Trigonometric Equations (Example) Ex 4: Find i) θ, 0° ≤ θ < 360° ii) all degree solutions – use a calculator to estimate: a) b) c) 12

13. Solving Quadratic Trigonometric Equations

14. Solving Quadratic Equations 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 14

15. Factoring a Quadratic To attempt factoring : – Always look for a GCF (greatest common factor) If present, factoring out the GCF simplifies the problem – Find two numbers that multiply to a·c AND add to b Only using the coefficients (numbers) – If a = 1, we have an easy trinomial Can immediately write as two binomials – If a ≠ 1, we have a hard trinomial Expand the trinomial into four terms Use grouping Alternatively, can also use “Guess and Check” 15

16. Solving Quadratic Equations Using the Quadratic Formula An equation in the format can also be solved using the Quadratic Formula: To solve a quadratic equation using the Quadratic Formula: – Set one side of the quadratic equation to zero – Plug the values of a, b, and c into the Quadratic Formula a, b, and c are all NUMBERS – Simplify 16

17. Solving Quadratic Trigonometric Equations Solving quadratic trigonometric equations is very similar EXCEPT we: – Attempt to factor or use the Quadratic Formula on a trigonometric function 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 – Be aware of extraneous solutions if fractions OR functions other than sine or cosine enter into the equation 17

18. Solving Quadratic Trigonometric Equations (Example) Ex 5: Find i) x, 0 ≤ x < 2π ii) all radian solutions a) b) c) 18

19. Solving Quadratic Trigonometric Equations (Example) Ex 6: Find i) θ, 0° ≤ θ < 360° ii) all degree solutions – use a calculator to estimate: 19

20. Additional Examples Ex 7: In a) find all exact degree solutions and in b) find all exact radian solutions a) b) 20

21. Summary After studying these slides, you should be able to: – Solve Linear Trigonometric Equations – Solve Quadratic Trigonometric Equations Additional Practice – See the list of suggested problems for 6.1 Next lesson – More on Trigonometric Equations (Section 6.2) 21