Day 57 Solving Trigonometric Equations 5.3

1. Day 57Solving Trigonometric Equations 5.3

2. What You Should Learn • Use standard algebraic techniques to solve trigonometric equations. • Solve trigonometric equations of quadratic type. • Use inverse trigonometric functions to solve trigonometric equations.

3. Plan for the Day Review of Homework What does it mean to “solve” trigonometric equations? Finding the angle, [0,2π) Linear and Quadratic types - simplifying and factoring to solve. Homework 5.3 Page 376 # 1-15 odd, 22, 31

4. Introduction Your preliminary goal in solving a trigonometric equation is to isolate the trigonometric function in the equation. To solve a trigonometric equation, use standard algebraic techniques such as collecting like terms and factoring. For example, to solve the equation 2 sin x = 1, divide each side by 2 to obtain

5. Introduction • To solve for x, note in Figure 5.6 that the equation • has solutions x = /6 and x = 5/6 in the interval [0, 2). Figure 5.6

6. Introduction Moreover, because sin x has a period of 2, there are infinitely many other solutions We will talk about that later

7. Introduction Another way to show that the equation has infinitely many solutions is indicated below. Any angles that are coterminal with  /6 or 5 /6 will also be solutions of the equation. • When solving trigonometric equations, you should write your answer(s) using exact values rather than decimal approximations. Figure 5.7

8. Solving Trigonometric Equations Finding the angles that make the statement true. Several methods for solving these types of equations that are similar to methods used with polynomials. • Combining like terms • Taking square roots • Factoring Since there are an infinite number of answers, the values of the variable were given as an expression in terms of the period of the function.

9. Using inverse functions to solve Using the unit circle to get exact values, solve for x, [0, 2π): • cos x = ½ • sin x = 0 • tan x = ±3 • cos x = -1

10. Using inverse functions to solve Using the unit circle to get exact values, solve for x, [0, 2π): • cos x = ½ {π/3, 5π/3} • sin x = 0 {0, π} • tan x = ±3 {π/3, 2π/3, 4π/3, 5π/3} • cos x = -1 {π}

11. Many times you will have to manipulate the equation to solve • Combining like terms • Using identities to simplify • Factoring

12. Try This… Solve for x: 0 = 4x – 2 0 = 4sin x – 2

13. Try This… Solve for x: 0 = x2 – 3 0 = tan2 x - 3

14. Try This… Solve for x: 0 = 4x2 + 2x 0 = 4 cos2 x + 2 cos x

15. Try This… Solve for x: 0 = 2x2 + 3x + 1 0 = 2cos2 x + 3cos x + 1

16. Solve 4sin2 x– 3 = 0 sin x – cos x sin x = 0 sin x (sec2 x + 1) = 0 2sin2 x= 2 + cos x

17. A couple of rules… • Look for values on the unit circle • When taking the square root, don’t forget the + and – • Never divide by a variable , move to the other side of the equation and factor! Given 3tan3 x = tan x Subtract tan from both sides 3tan3 x – tan x = 0 Factor tan x(3tan2 x – 1) = 0 Solve: tan x = 0 or 3tan2 x – 1 = 0 tan2 x = 1/3

18. Homework 31 5.3 Page 376 # 1-15 odd, 22, 31 Find all the solutions from [0, 2π)