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Trigonometric Equations

Trigonometric Equations. Solve Equations Involving a Single Trig Function. Checking if a Number is a Solution. Finding All Solutions of A Trig Equation.

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Trigonometric Equations

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  1. Trigonometric Equations Solve Equations Involving a Single Trig Function

  2. Checking if a Number is a Solution

  3. Finding All Solutions of A Trig Equation • Remember, trigonometric functions are periodic. Therefore, there an infinite number of solutions to the equation. To list all of the answers, we will have to determine a formula.

  4. Finding All Solutions of A Trig Equation • Tan q = 1 • tan-1(tan q) = tan-1 (1) • q = p/4 • To find all of the solutions, we need to remember that the period of the tangent function is p. • Therefore, the formula for all of the solutions is

  5. Finding All Solutions of A Trig Equation • cos q = 0 • cos-1 (cos q) = cos-1 0 • q = 0 • The period for cos is 2p. Therefore, the formula for all answers is 0 ± 2pk (k is an integer).

  6. Finding All Solutions of A Trig Equation

  7. Solving a Linear Trig Equation • Solve

  8. Solving a Trig Equation • Solve the equation on the interval 0 ≤ θ ≤ 2p

  9. Solving a Trig Equation • Solve the equation on the interval 0 ≤ θ ≤ 2p

  10. Solving a Trig Equation

  11. Solving a Trig Equation • The number of answers to a trig equation on the interval 0 ≤ θ ≤ 2p will be double the number in front of θ. In other words, if the angle is 2 θ the number of answers is 4. If the angle is 3 θ the number of answers is 6. If the angle is 4 θ the number of answers is 8, etc. unless the answer is a quadrantal angle.

  12. Solving a Trig Equation • Keep adding 2p to the answers until you have the needed angles.

  13. Solving a Trig Equation • Solve the equation on the interval 0 ≤ θ ≤ 2p

  14. Solving a Trig Equation

  15. Solving a Trig Equation • Solve the equation on the interval 0 ≤ θ ≤ 2p

  16. Solving a Trig Equation with a Calculator • sin θ = 0.4 • sin-1 (sin θ) = sin-1 0.4 • θ = .411, p - .411 = 2.73 • sec θ = -4 1/cos θ = -4 cos θ = -¼ • cos-1 (cos θ) = cos-1 (-¼) θ = 1.82 • Need to find reference angle because this is a quadrant II answer.

  17. Solving a Trig Equation with a Calculator • To find reference angle given a Quad II angle • p – answer (p – 1.82 = 1.32) • Now add p to this answer (p + 1.32) • θ = 4.46

  18. Snell’s Law of Refraction • Light, sound and other waves travel at different speeds, depending on the media (air, water, wood and so on) through which they pass. Suppose that light travels from a point A in one medium, where its speed is v1, to a point B in another medium, where its speed is v2. Angle θ1 is called the angle of incidence and the angle θ2 is the angle of refraction.

  19. Snell’s Law of Refraction • Snell’s Law states that

  20. Snell’s Law of Refraction Some indices of refraction are given in the table on page 512

  21. Snell’s Law of Refraction • The index of refraction of light in passing from a vacuum into water is 1.33. If the angle of incidence is 40o, determine the angle of refraction.

  22. Snell’s Law of Refraction

  23. Solving Trig Equations • Tutorial • Sample Problems • Video Explanations

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