5 3 solving trig eqns
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To solve trigonometric equations: 1) Use standard algebraic techniques such as collecting like terms and factoring to isolate the trigonometric function. 2) Many equations are of quadratic type (ax^2 + bx + c = 0) and can be solved by factoring or using the Quadratic Formula. 3) Equations involving multiple angles (sin ku, cos ku) are first solved for ku, then divided by k. 4) Inverse trigonometric functions can be used to solve equations, with the domain restrictions of the inverse functions considered.
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Example 9 – Solution
Setting each factor equal to zero, you obtain two solutions
in the interval (–/2, /2). [Recall that the range of the
inverse tangent function is (–/2, /2).]
tan x – 3 = 0 and tan x + 1 = 0
tan x = 3 tan x = –1
x = arctan 3
cont’d](https://image.slidesharecdn.com/53solvingtrigeqns-140816220913-phpapp01/85/5-3-solving-trig-eqns-21-320.jpg)
5 3 solving trig eqns
- 1. 5.3 SOLVING TRIGONOMETRIC EQUATIONS Copyright © Cengage Learning. All rights reserved.
- 2. 2 • Use standard algebraic techniques to solve trigonometric equations. • Solve trigonometric equations of quadratic type. • Solve trigonometric equations involving multiple angles. • Use inverse trigonometric functions to solve trigonometric equations. What You Should Learn
- 4. 4 Introduction To solve a trigonometric equation, use standard algebraic techniques such as collecting like terms and factoring. Your preliminary goal in solving a trigonometric equation is to isolate the trigonometric function in the equation. For example, to solve the equation 2 sin x = 1, divide each side by 2 to obtain
- 5. 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. 6 Introduction Moreover, because sin x has a period of 2, there are infinitely many other solutions, which can be written as and where n is an integer, as shown in Figure 5.6. General solution
- 7. 7 Introduction Another way to show that the equation has infinitely many solutions is indicated in Figure 5.7. 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. 8 Example 1 – Collecting Like Terms Solve Solution: Begin by rewriting the equation so that sin x is isolated on one side of the equation. Write original equation. Add sin x to each side. Subtract from each side.
- 9. 9 Example 1 – Solution Because sin x has a period of 2, first find all solutions in the interval [0, 2). These solutions are x = 5/4 and x = 7/4. Finally, add multiples of 2 to each of these solutions to get the general form and where n is an integer. cont’d Combine like terms. Divide each side by 2. General solution
- 10. 10 Equations of Quadratic Type
- 11. 11 Equations of Quadratic Type Many trigonometric equations are of quadratic type ax2 + bx + c = 0. Here are a couple of examples. Quadratic in sin x Quadratic in sec x 2 sin2 x – sin x – 1 = 0 sec2 x – 3 sec x – 2 = 0 2(sin x)2 – sin x – 1 = 0 (sec x)2 – 3(sec x) – 2 = 0 To solve equations of this type, factor the quadratic or, if this is not possible, use the Quadratic Formula.
- 12. 12 Example 4 – Factoring an Equation of Quadratic Type Find all solutions of 2 sin2 x – sin x – 1 = 0 in the interval [0, 2). Solution: Begin by treating the equation as a quadratic in sin x and factoring. 2 sin2 x – sin x – 1 = 0 (2 sin x + 1)(sin x – 1) = 0 Write original equation. Factor.
- 13. 13 Example 4 – Solution Setting each factor equal to zero, you obtain the following solutions in the interval [0, 2). 2 sin x + 1 = 0 and sin x – 1 = 0 cont’d
- 14. 14 Functions Involving Multiple Angles
- 15. 15 Functions Involving Multiple Angles The next example involves trigonometric functions of multiple angles of the forms sin ku and cos ku. To solve equations of these forms, first solve the equation for ku, then divide your result by k.
- 16. 16 Example 7 – Functions of Multiple Angles Solve 2 cos 3t – 1 = 0. Solution: 2 cos 3t – 1 = 0 2 cos 3t = 1 Write original equation. Add 1 to each side. Divide each side by 2.
- 17. 17 Example 7 – Solution In the interval [0, 2), you know that 3t = /3 and 3t = 5/3 are the only solutions, so, in general, you have and Dividing these results by 3, you obtain the general solution and where n is an integer. cont’d General solution
- 19. 19 Using Inverse Functions In the next example, you will see how inverse trigonometric functions can be used to solve an equation.
- 20. 20 Example 9 – Using Inverse Functions Solve sec2 x – 2 tan x = 4. Solution: sec2 x – 2 tan x = 4 1 + tan2 x – 2 tan x – 4 = 0 tan2 x – 2 tan x – 3 = 0 (tan x – 3)(tan x + 1) = 0 Write original equation. Pythagorean identity Combine like terms. Factor.
- 21. 21 Example 9 – Solution Setting each factor equal to zero, you obtain two solutions in the interval (–/2, /2). [Recall that the range of the inverse tangent function is (–/2, /2).] tan x – 3 = 0 and tan x + 1 = 0 tan x = 3 tan x = –1 x = arctan 3 cont’d
- 22. 22 Example 9 – Solution Finally, because tan x has a period of , you obtain the general solution by adding multiples of x = arctan 3 + n and where n is an integer. You can use a calculator to approximate the value of arctan 3. cont’d General solution