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Chapter 6 6.5 Trigonometric equations

Find all solutions of a trigonometric equation. Solve equations with multiple angles. Solve trigonometric equations quadratic in form. Use factoring to separate different functions in trigonometric equations. Use identities to solve trigonometric equations.

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Chapter 6 6.5 Trigonometric equations

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  1. Find all solutions of a trigonometric equation. • Solve equations with multiple angles. • Solve trigonometric equations quadratic in form. • Use factoring to separate different functions in trigonometric equations. • Use identities to solve trigonometric equations. • Use a calculator to solve trigonometric equations Chapter 66.5 Trigonometric equations Dr .Hayk Melikyan Departmen of Mathematics and CS melikyan@nccu.edu

  2. Trigonometric Equations and Their Solutions A trigonometric equation is an equation that contains a trigonometric expression with a variable, such as sin x. The values that satisfy such an equation are its solutions. (There are trigonometric equations that have no solution.) When an equation includes multiple angles, the period of the function plays an important role in ensuring that we do not leave out any solutions.

  3. Equations Involving a Single Trigonometric Function • To solve an equation containing a single trigonometric function: • • Isolate the function on one side of the equation. • sinx = a (-1 ≤ a ≤ 1 ) • cosx = a (-1 ≤ a ≤ 1 ) • tan x = a ( for any real a ) • • Solve for the variable.

  4. Trigonometric Equations x y y = cos x 1 y = 0.5 x –4  –2  2  4  –1 cos x = 0.5 has infinitely many solutions for –< x <  y y = cos x 1 0.5 2  cos x = 0.5 has two solutions for 0 < x < 2 –1

  5. This is the given equation. 3 sin x- 2 = 5 sin x- 1 Subtract 5 sin x from both sides. 3 sin x- 5 sin x- 2 = 5 sin x- 5 sin x – 1 Simplify. -2 sin x- 2 =-1 Add 2 to both sides. -2 sin x= 1 Divide both sides by -2 and solve for sin x. sin x= -1/2 Text Example Solve the equation: 3 sin x- 2 = 5 sin x- 1. Solution The equation contains a single trigonometric function, sin x. Step 1Isolate the function on one side of the equation. We can solve for sin x by collecting all terms with sin x on the left side, and all the constant terms on the right side.

  6. Solution The given equation is in quadratic form 2t2+t- 1 = 0 with t= cos x. Let us attempt to solve the equation using factoring. This is the given equation. 2 cos2x+ cos x- 1 = 0 Factor. Notice that 2t2 + t – 1 factors as (t – 1)(2t + 1). (2 cos x- 1)(cos x+ 1) = 0 Set each factor equal to 0. 2 cos x- 1= 0 or cos x+ 1 = 0 2 cos x= 1 cos x= -1 Solve for cos x. Text Example Solve the equation: 2 cos2 x+ cos x- 1 = 0, 0 £x< 2p. cos x= 1/2 x=px= 2pppx=p The solutions in the interval [0, 2p) are p/3, p, and 5p/3.

  7. Example • Solve the following equation: Solution:

  8. Example • Solve the equation on the interval [0,2) Solution:

  9. Example Solve the equation on the interval [0,2) Solution:

  10. Example • Solve the equation on the interval [0,2) Solution:

  11. Example: Finding all Solutions of a Trigonometric Equation • Solve the equation: • Step 1 Isolate the function on one side of the equation.

  12. Example: Finding all Solutions of a Trigonometric Equation (continued) • Solve the equation: • Step 2 Solve for the variable. Solutions for this equation in are: The solutions for this equation are:

  13. Solving an Equation with a Multiple Angle • Solve the equation: Because the period is all solutions for this equation are given by

  14. Solving an Equation with a Multiple Angle Because the period is all solutions for this equation are given by • Solve the equation: In the interval , the solutions are:

  15. Solving a Trigonometric Equation Quadratic in Form • Solve the equation: The solutions in the interval for this equation are:

  16. Using Factoring to Separate Different Functions • Solve the equation: The solutions for this equation in the interval are:

  17. Using an Identity to Solve a Trigonometric Equation • Solve the equation: The solutions in the interval are

  18. Solving Trigonometric Equations with a Calculator tanx is positive in quadrants I and III • Solve the equation, correct to four decimal places, for In quadrant I In quadrant III The solutions for this equation are 1.2592 and 4.4008.

  19. Using a Calculator to Solve Trigonometric Equations Sin x is negative in quadrants III and IV • Solve the equation, correct to four decimal places, for In quadrant III In quadrant IV The solutions for this equation are 3.3752 and 6.0496.

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