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10.4.2: Homework

  • Page ID
    197620
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    Reading Questions

    1. What is the ultimate goal when solving a trigonometric equation, according to the text?
    2. When an equation contains a squared trigonometric function, what type of identity is often useful for simplifying it?
    3. When solving an equation like \(3\cos(x)\sin^2(x) = \sin^2(x)\), what is the recommended first step instead of dividing by \(\sin^2(x)\)?
    4. What algebraic principle allows you to solve a factored trigonometric equation like \(\sin(A)[2\sin(A)+1] = 0\)?
    5. If an equation contains trigonometric functions with different arguments, like \(\cos(2x)\) and \(\cos(x)\), what is the general strategy?
    6. If an equation contains a sum or difference of trigonometric functions that cannot be easily factored, like \(\cos(3x) - \cos(5x) = 0\), what type of identity might be required?
    7. Under what circumstances must you check your solutions for extraneous solutions?
    8. An expression of the form \(m\sin(Bx) + n\cos(Bx)\) can be rewritten as a single sinusoidal function. What is the amplitude of this new function?

    Homework

    Concept Check

    1. Will there always be solutions to trigonometric function equations? If not, describe an equation that would not have a solution. Explain why or why not.

    2. When solving a trigonometric equation involving more than one trigonometric function, do we always want to try to rewrite it so it is expressed in one trigonometric function? Why or why not?

    Basic Skills

    For the following exercises, solve the equation, giving exact solutions on the interval \( \left[ 0,2\pi \right) \).

    1. \(\sin\left( 2\theta \right)+\sqrt{2} \cos\left( \theta \right)=0\)

    2. \(\sin\left( 2\alpha \right) \sin\left( \alpha \right)=\cos\left( \alpha \right)\)

    3. \(\cos\left( 2t \right)-5 \cos\left( t \right)+3=0\)

    4. \(\cos\left( 2x \right)+3 \sin\left( x \right)=2\)

    5. \(\tan\left( 2\beta \right)+2 \sin\left( \beta \right)=0\)

    6. \(\tan\left( 2z \right)-2 \cos\left( z \right)=0\)

    7. \(3 \cos\left( \phi \right)-\sin \left(\frac{\pi}{2}-\phi\right)=\sqrt{3}\)

    8. \(\sin\left( w \right)+\cos \left(\frac{\pi}{2}-w\right)=1\)

    9. \(\sin\left( 2\phi \right) \cos\left( \phi \right)+\cos\left( 2\phi \right) \sin\left( \phi \right)=1\)

    10. \(\cos\left( \theta \right) \cos\left( 3\theta \right)+\sin\left( \theta \right) \sin\left( 3\theta \right)=\frac{\sqrt{2}}{2}\)

    11. \(\sin \left( x \right) = \cos \left( x \right)\)

    12. \(\sin \left( 2x \right) = \sin \left( x \right)\)

    13. \(\sin \left( 2x \right) = \cos \left( x \right)\)

    14. \(2\tan^2\left( t \right)=3\sec\left( t \right)\)

    15. \(\cos \left( 2x \right) = \sin \left( x \right)\)

    16. \(\cos \left( 2x \right) = \cos \left( x \right)\)

    17. \(\cos\left( 2x \right) = 2 - 5\cos\left( x \right)\)

    18. \(3\cos\left( 2x \right) + \cos\left( x \right) + 2 = 0\)

    19. \(\cos\left( 2x \right) = 5\sin\left( x \right) - 2\)

    20. \(3\cos\left( 2x \right) = \sin\left( x \right) + 2\)

    21. \(2\sec^2\left( x \right) = 3 - \tan\left( x \right)\)

    22. \(\tan^2\left( x \right) = 1-\sec\left( x \right)\)

    23. \(\cot^2\left( x \right) = 3\csc\left( x \right) - 3\)

    24. \(\sec\left( x \right) = 2\csc\left( x \right)\)

    25. \(\cos\left( x \right)\csc\left( x \right)\cot\left( x \right) = 6-\cot^2\left( x \right)\)

    26. \(\sin\left( 2x \right) = \tan\left( x \right)\)

    27. \(\cot^4\left( x \right) = 4\csc^2\left( x \right) - 7\)

    28. \(\cos\left( 2x \right) + \csc^2\left( x \right) = 0\)

    29. \(\tan^{3} \left( x \right) = 3\tan \left( x \right)\)

    30. \(\cos\left( 6x \right)-\cos\left( 3x \right)=0\)

    31. \(\sin^2\left( x \right)-\cos^2\left( x \right)-\sin\left( x \right)=0\)

    32. \(\tan^{2} \left( x \right) = \frac{3}{2} \sec \left( x \right)\)

    33. \(2\sin\left( x \right)\cos\left( x \right)-\sin\left( x \right)+2\cos\left( x \right)-1=0\)

    34. \(\cos^{3} \left( x \right) = -\cos \left( x \right)\)

    35. \(\tan\left( 2x \right) - 2\cos\left( x \right) = 0\)

    36. \(\csc^3\left( x \right) + \csc^2\left( x \right) = 4\csc\left( x \right) + 4\)

    37. \(2\tan\left( x \right) = -1 - \tan^2\left( x \right)\)

    38. \(\tan \left( x \right) = \sec \left( x \right)\)

    39. \(\sin\left( 6x \right) \cos\left( x \right) = -\cos\left( 6x \right) \sin\left( x \right)\)

    40. \(\sin\left( 3x \right)\cos\left( x \right) = \cos\left( 3x \right) \sin\left( x \right)\)

    41. \(\cos\left( 2x \right)\cos\left( x \right) + \sin\left( 2x \right)\sin\left( x \right) = 1\)

    42. \(\cos\left( 5x \right)\cos\left( 3x \right) - \sin\left( 5x \right)\sin\left( 3x \right) = \frac{\sqrt{3}}{2}\)

    43. \(\sin\left( x \right) + \cos\left( x \right) = 1\)

    44. \(\sin\left( x \right) + \sqrt{3} \cos\left( x \right) = 1\)

    45. \(\sqrt{2} \cos\left( x \right) - \sqrt{2} \sin\left( x \right) = 1\)

    46. \(\sqrt{3} \sin\left( 2x \right) + \cos\left( 2x \right) = 1\)

    47. \(\cos\left( 2x \right) - \sqrt{3} \sin\left( 2x \right) = \sqrt{2}\)

    48. \(9\cos\left( 2\theta \right)=9\cos^2\left( \theta \right) -4\)

    49. \(3\sqrt{3}\sin\left( 3x \right) - 3\cos\left( 3x \right) = 3\sqrt{3}\)

    50. \(\cos\left( 3x \right) = \cos\left( 5x \right)\)

    51. \(\cos\left( 4x \right) = \cos\left( 2x \right)\)

    52. \(\sin\left( 5x \right) = \sin\left( 3x \right)\)

    53. \(\cos\left( 5x \right) = -\cos\left( 2x \right)\)

    54. \(\sin\left( 6x \right) + \sin\left( x \right) = 0\)

    55. \(\tan\left( x \right) = \cos\left( x \right)\)

    For the following exercises, solve the equation for \(0^{\circ} \leq \theta \leq 360^{\circ}\). Round angles to three decimal places if necessary.

    1. \(\cos\left( \theta \right)-\sin^2\left( \theta \right)+1=0\)

    2. \(4 \sin\left( \theta \right)+2 \cos^2\left( \theta \right)-3=-1\)

    3. \(1-\sin\left( \theta \right)-2 \cos^2\left( \theta \right)=0\)

    4. \(3 \cos^2\left( \theta \right)-\sin^2\left( \theta \right)=2\)

    5. \(2 \cos\left( \theta \right) \tan\left( \theta \right)+1=0\)

    6. \(\cos\left( \theta \right)-\sin\left( \theta \right)=0\)

    7. \(\frac{1}{3} \cos\left( \theta \right)=\sin\left( \theta \right)\)

    8. \(5 \sin\left( C \right)=2 \cos\left( C \right)\)

    For the following exercises, find all radian solutions to the equation.

    1. \(\sin \left(3x\right)\cos \left(6x\right)-\cos \left(3x\right)\sin \left(6x\right)= -0.9\)

    2. \(\sin \left(6x\right)\cos \left(11x\right)-\cos \left(6x\right)\sin \left(11x\right)= -0.1\)

    3. \(\cos \left(2x\right)\cos \left(x\right)+\sin \left(2x\right)\sin \left(x\right)=1\)

    4. \(\cos \left(5x\right)\cos \left(3x\right)-\sin \left(5x\right)\sin \left(3x\right)=\frac{\sqrt{3} }{2}\)

    5. \(6\sin\left( 2t \right)+9\sin\left( t \right)=0\)

    For the following exercises, find all radian solutions to the equation.

    1. \(\cos \left(5x\right)=-\cos \left(2x\right)\)

    2. \(\sin \left(5x\right)=\sin \left(3x\right)\)

    3. \(\cos \left(6\theta \right)-\cos \left(2\theta \right)=\sin \left(4\theta \right)\)

    4. \(\cos \left(8\theta \right)-\cos \left(2\theta \right)=\sin \left(5\theta \right)\)

    For the following exercises, rewrite the expression as a single function of the form \(A\,\sin\left(Bx+\phi\right)\).

    1. \(4\sin \left(x\right)-6\cos \left(x\right)\)

    2. \(-\sin \left(x\right)-5\cos \left(x\right)\)

    3. \(5\sin \left(3x\right)+2\cos \left(3x\right)\)

    4. \(-3\sin \left(5x\right)+4\cos \left(5x\right)\)

    For the following exercises, find the first two positive (radian) solutions.

    1. \(-5\sin \left(x\right)+3\cos \left(x\right)=1\)

    2. \(3\sin \left(x\right)+\cos \left(x\right)=2\)

    3. \(3\sin \left(2x\right)-5\cos \left(2x\right)=3\)

    4. \(-3\sin \left(4x\right)-2\cos \left(4x\right)=1\)

    For the following exercises, solve the equation for \(0^{\circ} \leq x \lt 360^{\circ}\).

    1. \( 9 \sin^2\left( \theta \right)-6 \sin\left( \theta \right)=1\)

    2. \( 4 \cos^2\left( \theta \right)+4 \cos\left( \theta \right)=1\)

    3. \( \sec^2\left( \alpha \right)-2 \sec\left( \alpha \right)-3=0\)

    4. \( \csc^2\left( \beta \right)+4 \csc\left( \beta \right)-10=0\)

    5. \( \csc^2\left( x \right)+4 \csc\left( x \right)-7=0\)

    6. \( 3 \cot^2\left( x \right)-3 \cot\left( x \right)-1=0\)

    7. \( 2 \sin^2\left( x \right)=1-\cos\left( x \right)\)

    8. \( \cos^2\left( \alpha \right)+4=2 \sin\left( \alpha \right)-3\)

    9. \( \cos^2\left( \beta \right)-3 \sin\left( \beta \right)+2 \sin^2\left( \beta \right)=0\)

    10. \( \sin^2\left( \theta \right)=2 \cos\left( \theta \right)+3 \cos^2\left( \theta \right)\)

    11. \( \sec^2\left( x \right)=2 \tan\left( x \right)+4\)

    12. \( 3 \tan^2\left( x \right)=\sec\left( x \right)+2\)

    13. \( \cos\left( \alpha \right)+1=2 \cos\left( 2\alpha \right)\)

    14. \( \cos\left( 2x \right)-3 \sin\left( x \right)-2=0\)

    15. \( \csc^2\left( \theta \right)=\cot\left( \theta \right)+5\)

    16. \( \csc\left( \theta \right)+5=2 \cot^2\left( \theta \right)+2\)


    This page titled 10.4.2: Homework was last modified on Tue, 08 Jul 2025 17:46:28 GMT and is shared under a not declared license and was authored, remixed, and/or curated by Roy Simpson.

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