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# Week 6 - Trigonometry

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### Week 6 - Trigonometry

1. 1. Day 261. Opener.Solve for x: x 1. 10 = 5.71 3x 2. 7e = 312
2. 2. 2. Exponential and Logarithmic Equations.Exponential equations: An exponential equation is an equationcontaining a variable in an exponent.Logarithmic equations: Logarithmic equations containlogarithmic expressions and constants.Property of Logarithms, part 2: ≠ 1, thenIf x, y and a are positive numbers, aIf x = y, then log a x = log a y
3. 3. 2. Exponential and Logarithmic Equations. x+2 2 x+1Example: Solve 2 =3
4. 4. 2. Exponential and Logarithmic Equations. x+2 2 x+1Example: Solve 2 =3 x+2 2 x+1ln 2 = ln 3
5. 5. 2. Exponential and Logarithmic Equations. x+2 2 x+1Example: Solve 2 =3 x+2 2 x+1 Take the log of both sidesln 2 = ln 3
6. 6. 2. Exponential and Logarithmic Equations. x+2 2 x+1 Example: Solve 2 =3 x+2 2 x+1 Take the log of both sides ln 2 = ln 3( x + 2 ) ln 2 = ( 2x + 1) ln 3
7. 7. 2. Exponential and Logarithmic Equations. x+2 2 x+1 Example: Solve 2 =3 x+2 2 x+1 Take the log of both sides ln 2 = ln 3( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithms
8. 8. 2. Exponential and Logarithmic Equations. x+2 2 x+1 Example: Solve 2 =3 x+2 2 x+1 Take the log of both sides ln 2 = ln 3( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithmsx ln 2 + 2 ln 2 = 2x ln 3 + ln 3
9. 9. 2. Exponential and Logarithmic Equations. x+2 2 x+1 Example: Solve 2 =3 x+2 2 x+1 Take the log of both sides ln 2 = ln 3( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithmsx ln 2 + 2 ln 2 = 2x ln 3 + ln 3 Distributive property
10. 10. 2. Exponential and Logarithmic Equations. x+2 2 x+1 Example: Solve 2 =3 x+2 2 x+1 Take the log of both sides ln 2 = ln 3( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithmsx ln 2 + 2 ln 2 = 2x ln 3 + ln 3 Distributive propertyx ln 2 − 2x ln 3 = ln 3 − 2 ln 2
11. 11. 2. Exponential and Logarithmic Equations. x+2 2 x+1 Example: Solve 2 =3 x+2 2 x+1 Take the log of both sides ln 2 = ln 3( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithmsx ln 2 + 2 ln 2 = 2x ln 3 + ln 3 Distributive property Isolate terms (variable onx ln 2 − 2x ln 3 = ln 3 − 2 ln 2 one side of the equation).
12. 12. 2. Exponential and Logarithmic Equations. x+2 2 x+1 Example: Solve 2 =3 x+2 2 x+1 Take the log of both sides ln 2 = ln 3( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithmsx ln 2 + 2 ln 2 = 2x ln 3 + ln 3 Distributive property Isolate terms (variable onx ln 2 − 2x ln 3 = ln 3 − 2 ln 2 one side of the equation).x ( ln 2 − 2 ln 3) = ln 3 − 2 ln 2
13. 13. 2. Exponential and Logarithmic Equations. x+2 2 x+1 Example: Solve 2 =3 x+2 2 x+1 Take the log of both sides ln 2 = ln 3( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithmsx ln 2 + 2 ln 2 = 2x ln 3 + ln 3 Distributive property Isolate terms (variable onx ln 2 − 2x ln 3 = ln 3 − 2 ln 2 one side of the equation).x ( ln 2 − 2 ln 3) = ln 3 − 2 ln 2 Common factor, x.
14. 14. 2. Exponential and Logarithmic Equations. x+2 2 x+1 Example: Solve 2 =3 x+2 2 x+1 Take the log of both sides ln 2 = ln 3( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithmsx ln 2 + 2 ln 2 = 2x ln 3 + ln 3 Distributive property Isolate terms (variable onx ln 2 − 2x ln 3 = ln 3 − 2 ln 2 one side of the equation).x ( ln 2 − 2 ln 3) = ln 3 − 2 ln 2 Common factor, x. ln 3 − 2 ln 2x= ln 2 − 2 ln 3
15. 15. 2. Exponential and Logarithmic Equations. x+2 2 x+1 Example: Solve 2 =3 x+2 2 x+1 Take the log of both sides ln 2 = ln 3( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithmsx ln 2 + 2 ln 2 = 2x ln 3 + ln 3 Distributive property Isolate terms (variable onx ln 2 − 2x ln 3 = ln 3 − 2 ln 2 one side of the equation).x ( ln 2 − 2 ln 3) = ln 3 − 2 ln 2 Common factor, x. ln 3 − 2 ln 2x= Divide both sides by ln2 - 2ln3 ln 2 − 2 ln 3
16. 16. 2. Exponential and Logarithmic Equations.Example: Solve log 4 ( x + 3) = 2
17. 17. 2. Exponential and Logarithmic Equations.Example: Solve log 4 ( x + 3) = 2 2 4 = x+3
18. 18. 2. Exponential and Logarithmic Equations.Example: Solve log 4 ( x + 3) = 2 2 Deﬁnition of Logarithm 4 = x+3
19. 19. 2. Exponential and Logarithmic Equations.Example: Solve log 4 ( x + 3) = 2 2 Deﬁnition of Logarithm 4 = x+3 16 = x + 3
20. 20. 2. Exponential and Logarithmic Equations.Example: Solve log 4 ( x + 3) = 2 2 Deﬁnition of Logarithm 4 = x+3 16 = x + 3 Simplify
21. 21. 2. Exponential and Logarithmic Equations.Example: Solve log 4 ( x + 3) = 2 2 Deﬁnition of Logarithm 4 = x+3 16 = x + 3 Simplify 13 = x
22. 22. 2. Exponential and Logarithmic Equations.Example: Solve log 4 ( x + 3) = 2 2 Deﬁnition of Logarithm 4 = x+3 16 = x + 3 Simplify 13 = x Solve for x.
23. 23. 2. Exponential and Logarithmic Equations. Example: Solve log 4 ( x + 3) = 2 2 Deﬁnition of Logarithm 4 = x+3 16 = x + 3 Simplify 13 = x Solve for x.All solutions of Logarithmic equations must be checked, because negative numbers do not have logarithms.
24. 24. 2. Exponential and Logarithmic Equations.Example: Solve log 2 x + log 2 ( x − 7 ) = 3
25. 25. 2. Exponential and Logarithmic Equations.Example: Solve log 2 x + log 2 ( x − 7 ) = 3log 2 x ( x − 7 ) = 3
26. 26. 2. Exponential and Logarithmic Equations.Example: Solve log 2 x + log 2 ( x − 7 ) = 3log 2 x ( x − 7 ) = 3 Property of Logarithms
27. 27. 2. Exponential and Logarithmic Equations.Example: Solve log 2 x + log 2 ( x − 7 ) = 3log 2 x ( x − 7 ) = 3 Property of Logarithms 3 2 = x ( x − 7)
28. 28. 2. Exponential and Logarithmic Equations.Example: Solve log 2 x + log 2 ( x − 7 ) = 3log 2 x ( x − 7 ) = 3 Property of Logarithms 3 2 = x ( x − 7) Deﬁnition of Logarithm
29. 29. 2. Exponential and Logarithmic Equations.Example: Solve log 2 x + log 2 ( x − 7 ) = 3log 2 x ( x − 7 ) = 3 Property of Logarithms 3 2 = x ( x − 7) Deﬁnition of Logarithm 2 8 = x − 7x
30. 30. 2. Exponential and Logarithmic Equations.Example: Solve log 2 x + log 2 ( x − 7 ) = 3log 2 x ( x − 7 ) = 3 Property of Logarithms 3 2 = x ( x − 7) Deﬁnition of Logarithm 2 8 = x − 7x Simplify
31. 31. 2. Exponential and Logarithmic Equations.Example: Solve log 2 x + log 2 ( x − 7 ) = 3log 2 x ( x − 7 ) = 3 Property of Logarithms 3 2 = x ( x − 7) Deﬁnition of Logarithm 2 8 = x − 7x Simplify 2 0 = x − 7x − 8
32. 32. 2. Exponential and Logarithmic Equations.Example: Solve log 2 x + log 2 ( x − 7 ) = 3log 2 x ( x − 7 ) = 3 Property of Logarithms 3 2 = x ( x − 7) Deﬁnition of Logarithm 2 8 = x − 7x Simplify 2 Write cuadratic equation in 0 = x − 7x − 8 standard form
33. 33. 2. Exponential and Logarithmic Equations.Example: Solve log 2 x + log 2 ( x − 7 ) = 3log 2 x ( x − 7 ) = 3 Property of Logarithms 3 2 = x ( x − 7) Deﬁnition of Logarithm 2 8 = x − 7x Simplify 2 Write cuadratic equation in 0 = x − 7x − 8 standard form0 = ( x − 8 ) ( x + 1)
34. 34. 2. Exponential and Logarithmic Equations.Example: Solve log 2 x + log 2 ( x − 7 ) = 3log 2 x ( x − 7 ) = 3 Property of Logarithms 3 2 = x ( x − 7) Deﬁnition of Logarithm 2 8 = x − 7x Simplify 2 Write cuadratic equation in 0 = x − 7x − 8 standard form0 = ( x − 8 ) ( x + 1) Solve by factoring
35. 35. 2. Exponential and Logarithmic Equations.Example: Solve log 2 x + log 2 ( x − 7 ) = 3log 2 x ( x − 7 ) = 3 Property of Logarithms 3 2 = x ( x − 7) Deﬁnition of Logarithm 2 8 = x − 7x Simplify 2 Write cuadratic equation in 0 = x − 7x − 8 standard form0 = ( x − 8 ) ( x + 1) Solve by factoring x = 8 or x = -1
36. 36. 2. Exponential and Logarithmic Equations.Example: Solve log 2 x + log 2 ( x − 7 ) = 3log 2 x ( x − 7 ) = 3 Property of Logarithms 3 2 = x ( x − 7) Deﬁnition of Logarithm 2 8 = x − 7x Simplify 2 Write cuadratic equation in 0 = x − 7x − 8 standard form0 = ( x − 8 ) ( x + 1) Solve by factoring x = 8 or x = -1 Check!
37. 37. 2. Exponential and Logarithmic Equations.Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x
38. 38. 2. Exponential and Logarithmic Equations.Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x 4x − 3log ( 2x − 1) = log x
39. 39. 2. Exponential and Logarithmic Equations.Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x 4x − 3log ( 2x − 1) = log Property of Logarithms x
40. 40. 2. Exponential and Logarithmic Equations.Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x 4x − 3log ( 2x − 1) = log Property of Logarithms x 4x − 3 2x − 1 = x
41. 41. 2. Exponential and Logarithmic Equations.Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x 4x − 3log ( 2x − 1) = log Property of Logarithms x 4x − 3 2x − 1 = Property of Logarithms x
42. 42. 2. Exponential and Logarithmic Equations.Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x 4x − 3log ( 2x − 1) = log Property of Logarithms x 4x − 3 2x − 1 = Property of Logarithms x Multiply both sides by x
43. 43. 2. Exponential and Logarithmic Equations.Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x 4x − 3log ( 2x − 1) = log Property of Logarithms x 4x − 3 2x − 1 = Property of Logarithms x x(2x - 1) = 4x - 3 Multiply both sides by x
44. 44. 2. Exponential and Logarithmic Equations.Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x 4x − 3log ( 2x − 1) = log Property of Logarithms x 4x − 3 2x − 1 = Property of Logarithms x x(2x - 1) = 4x - 3 Multiply both sides by x Distributive property
45. 45. 2. Exponential and Logarithmic Equations.Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x 4x − 3log ( 2x − 1) = log Property of Logarithms x 4x − 3 2x − 1 = Property of Logarithms x x(2x - 1) = 4x - 3 Multiply both sides by x 2x2 - x = 4x - 3 Distributive property
46. 46. 2. Exponential and Logarithmic Equations.Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x 4x − 3log ( 2x − 1) = log Property of Logarithms x 4x − 3 2x − 1 = Property of Logarithms x x(2x - 1) = 4x - 3 Multiply both sides by x 2x2 - x = 4x - 3 Distributive property 2x2 - 5x + 3 = 0
47. 47. 2. Exponential and Logarithmic Equations.Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x 4x − 3log ( 2x − 1) = log Property of Logarithms x 4x − 3 2x − 1 = Property of Logarithms x x(2x - 1) = 4x - 3 Multiply both sides by x 2x2 - x = 4x - 3 Distributive property Write quadratic equation in 2x2 - 5x + 3 = 0 standard form
48. 48. 2. Exponential and Logarithmic Equations.Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x 4x − 3log ( 2x − 1) = log Property of Logarithms x 4x − 3 2x − 1 = Property of Logarithms x x(2x - 1) = 4x - 3 Multiply both sides by x 2x2 - x = 4x - 3 Distributive property Write quadratic equation in 2x2 - 5x + 3 = 0 standard form (2x - 3)(x - 1) = 0
49. 49. 2. Exponential and Logarithmic Equations.Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x 4x − 3log ( 2x − 1) = log Property of Logarithms x 4x − 3 2x − 1 = Property of Logarithms x x(2x - 1) = 4x - 3 Multiply both sides by x 2x2 - x = 4x - 3 Distributive property Write quadratic equation in 2x2 - 5x + 3 = 0 standard form (2x - 3)(x - 1) = 0 Solve by factoring
50. 50. 2. Exponential and Logarithmic Equations.Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x 4x − 3log ( 2x − 1) = log Property of Logarithms x 4x − 3 2x − 1 = Property of Logarithms x x(2x - 1) = 4x - 3 Multiply both sides by x 2x2 - x = 4x - 3 Distributive property Write quadratic equation in 2x2 - 5x + 3 = 0 standard form (2x - 3)(x - 1) = 0 Solve by factoring x = 3/2 and x = 1
51. 51. 2. Exponential and Logarithmic Equations.Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x 4x − 3log ( 2x − 1) = log Property of Logarithms x 4x − 3 2x − 1 = Property of Logarithms x x(2x - 1) = 4x - 3 Multiply both sides by x 2x2 - x = 4x - 3 Distributive property Write quadratic equation in 2x2 - 5x + 3 = 0 standard form (2x - 3)(x - 1) = 0 Solve by factoring x = 3/2 and x = 1 Check!
52. 52. Day 27 1. Exercises.
53. 53. Day 271. Opener.
54. 54. Day 301. Quiz 4.1. “Quiz 4”.2. Name.3. Student Number.4. Date.
55. 55. 2. Quiz 4.1. How long does it take to double an investment of \$ 20,000.00 in abank paying an interest rate of 4% per year compounded monthly?Find the value of x in the following equations. Check your answers.2. 3x+4 = e5 x3. log12 ( x − 7 ) = 1− log12 ( x − 3)4. Character that maintained a robust dispute with Newton over thepriority of invention of calculus.5. How did Evariste Galois die, two days after leaving prison, at age21?6. Why isn’t there a Nobel Prize in mathematics?