10.3.2: Homework

Math 375: Pre-Calculus · LibreTexts

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    Reading Questions

    1. Why do trigonometric equations typically have an infinite number of solutions?
    2. What are the two fundamental concepts from trigonometry that are the keys to finding all solutions to a trigonometric equation?
    3. If you are solving \(\cos(\theta) = -0.6\), in which two quadrants will the solutions lie?
    4. After using a calculator to find a reference angle, how do you find the general solution for all angles coterminal with that solution?
    5. What does it mean for a trigonometric equation to be "quadratic-in-form"?
    6. What algebraic technique is used to solve an equation like \(2\cos^2(\theta) - \cos(\theta) - 1 = 0\)?
    7. When solving an equation involving \(\tan(x)\) or \(\cot(x)\), how does finding the general solution differ from finding it for sine or cosine?
    8. When solving an equation with a non-standard argument, such as \(\sin(2x) = 0.5\), what should you treat as the "angle" when finding the reference angle?
    9. How does the coefficient on the variable, like the '2' in \(\sin(2x)\), affect the number of solutions within the interval \([0, 2\pi]\)?
    10. Why does an equation like \(\sin(x) = 5\) have no solution?

    Skills Refresher

    Review the following skills you will need for this section.

    Skills Refresher

    For the following exercises, solve the equation.

    1. \((2x - 6)^2 = 4\)

    2. \(\sqrt{2x-6} = 4\)

    3. \(\dfrac{1}{2x-6} = 4\)

    4. \(|2x-6| = 4\)

    5. \(\log 2x-6 = 4\)

    6. \(2^{2x-6} = 4\)

    For the following exercises, the ”solution” contains an error. Find the error, and supply a correct solution.

    1.  \[\begin{array}{rrcl}
      & 3x^2-5x & = & 0 \\[6pt] \implies & 3x^2 & = & 5x \\[6pt] \implies & 3x & = & 5 \\[6pt] \implies & x & = & \dfrac{5}{3} \\[6pt] \end{array} \nonumber \]
    2.  \[\begin{array}{rrcl}
      & 4x^2 - 5x & = & 12 \\[6pt] \implies & x^2 & = & 3 \\[6pt] \implies & x & = & \sqrt{3} \\[6pt] \end{array} \nonumber \]
    3.  \[\begin{array}{rrcl}
      & (2x+1)^2 & = & 4 \\[6pt] \implies & 2x+1 & = & 2 \\[6pt] \implies & 2x & = & 1 \\[6pt] \implies & x & = & \dfrac{1}{2} \\[6pt] \end{array} \nonumber \]
    4.  \[\begin{array}{rrcl}
      & 2x^2-4x & = & 3 \\[6pt] \implies & 2x(x-2) & = & 3 \\[6pt] \implies & 2x=3 & \text{ or } & x-2=3 \\[6pt] \implies & x = \dfrac{3}{2} & \text{ or } & x=5 \\[6pt] \end{array}\nonumber \]
    Answers

    1. 2, 4

    2. 11

    3. \(\dfrac{25}{8}\)

    4. 1, 5

    5. 5003

    6. 4

    7. We can't divide by \(x\). \(x = 0, \dfrac{5}{3}\)

    8. 3 has two square roots. \(x = \pm \sqrt{3}\)

    9. 4 has two square roots. \(x = \dfrac{-3}{2},\dfrac{1}{2}\)

    10. One side must be zero to use the zero-factor principle. No real solutions.

    Homework

    Concept Check

    1. How many solutions between \(0^{\circ}\) and \(360^{\circ}\) does the equation \(\cos \left(\theta\right) = k\) have for each value of \(k\) between \(-1\) and \(0\)?

    2. How many solutions between \(0^{\circ}\) and \(360^{\circ}\) does the equation \(\cos \left(\theta\right) = k\) have for each value of \(k\) greater than \(1\)?

    3. How many solutions between \(0^{\circ}\) and \(360^{\circ}\) does the equation \(\sin^2 \left(\theta\right) = k\) have for each value of \(k\) between \(-1\) and \(0\)?

    4. If \(x=0.68\) is one solution of the equation \(\cos \left(x\right)=c\), what is other? Illustrate on a unit circle.

    5. If \(t=2.45\) is one solution of the equation \(\sin \left(t\right)=k\), what is the other? Illustrate on a unit circle.

    6. If \(\theta=1.73\) is one solution of the equation \(\tan \left(\theta\right)=C\), what is the other?

    7. Explain why the equation \(\cos \left(n x\right)=k, \quad 0 \lt k \lt 1\), has \(2 n\) solutions between \(0\) and \(2 \pi\).

    8. How many solutions are there to the equation \(\cos \left(\theta\right)=0.4\)? How many solutions are there between \(0\) and \(2\pi\)?

    9. Find all angles between \(0^{\circ}\) and \(360^{\circ}\) for which the statement is true.

      1. \(\cos \left(\theta\right) = -1\)

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

      3. \(\tan \left(\theta\right) = -1\)

    10.  

      1. Find two angles, \(0 \leq \theta \leq 360^{\circ}\), with \(\sin \left(\theta\right) = 0\).

      2. Find two angles, \(0 \leq \theta \leq 360^{\circ}\), with \(\cos \left(\theta\right) = 0\).

    11.  

      1. Find two angles, \(0 \leq \theta \leq 360^{\circ}\), with \(\sin \left(\theta\right) = \cos \left(\theta\right)\).

      2. Find two angles, \(0 \leq \theta \leq 360^{\circ}\), with \(\sin \left(\theta\right) = -\cos \left(\theta\right)\).

    Basic Skills

    For the following exercises,
    (a) Evaluate the expression at the given values of the variable.
    (b) Give one solution to the equation.

    1.  

      1. \(\sin\left(\theta\right)+\cos\left(\theta\right) ; \quad \theta=0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}\)

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

    2.  

      1. \(\sin ^2 \left(\alpha\right)-\cos \left(\alpha\right) ; \quad \alpha = \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi\)

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

    3.  

      1. \(\sin \left(\beta\right)+2 \cos ^2 \left(\beta\right) ; \quad \beta=210^{\circ}, 225^{\circ}, 240^{\circ}, 270^{\circ}\)

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

    4.  

      1. \(3 \cos ^2 \left(\phi\right)-\sin ^2 \left(\phi \right); \quad \phi=\frac{3\pi}{2}, \frac{5\pi}{3}, \frac{7\pi}{4}, \frac{11\pi}{6}\)

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

    For the following exercises, find all degree solutions (in exact form) and then state the solutions for \(0^{\circ} \leq \theta \leq 360^{\circ}\).

    1. \(7 \sin \left(\theta\right)+11=11\)

    2. \(3=5-4 \cos \left(\theta\right)\)

    3. \(3 \tan \left(\theta\right)=\sqrt{3}\)

    4. \(8 \csc \left(\theta\right)+5=-11\)

    5. \(9 \sec \left(\theta\right)+15=6\)

    6. \(6 \cot \left(\theta\right)+21=15\)

    7. \(0=\sqrt{2}+2 \sin \left(\theta\right)\)

    8. \(\sqrt{3} \cos \left(\theta\right)=-\frac{3}{2}\)

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

    10. \(1-\sin ^2 \left(\theta\right)=0\)

    11. \(3 \csc ^2 \left(\theta\right)-4=0\)

    12. \(0=4-2 \sec ^2 \left(\theta\right)\)

    13. \(1-\tan ^2 \left(\theta\right)=0\)

    14. \(0=2 \cot ^2 \left(\theta\right)-6\)

    15. \(2 \sin^2 \left(\theta\right)+\sin \left(\theta\right)-1=0\)

    16. \(4 - \cos\left( \theta \right) = 4 \cos^2 \left(\theta\right)+3\cos \left(\theta\right)+5\)

    17. \( 2\sin^2\left( \theta \right) - 11\sin\left( \theta \right) = -5 \)

    18. \( 2 = \csc^2\left( \theta \right) - \csc\left( \theta \right)\)

    19. \( \sec^2\left( \theta \right) + 3\sec\left( \theta \right) + 2 = 0\)

    20. \(\sin \left(4 x\right)=-1\)

    21. \(\cos \left(3 t\right)=0\)

    22. \(5 \tan \left(2 q\right)=0\)

    23. \(6 \sin \left(4 w\right)=-3 \sqrt{2}\)

    24. \(4 \cos \left(3 \phi\right)=-2\)

    For the following exercises, find all radian solutions (in exact form) and then state the solutions for \( 0 \leq \theta \leq 2\pi \).

    1. \(\sqrt{2} \sin \left(\theta\right)+9=8\)

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

    3. \(13=13-4 \cos \left(\theta\right)\)

    4. \(4 \tan \left(\theta\right) - 2 = -6\)

    5. \(8 \csc \left(\theta\right)+5=-3\)

    6. \(2\cos^2\left( t \right)+\cos\left( t \right)=1\)

    7. \(\sec\left( x \right)\sin\left( x \right)-2\sin\left( x \right)=0\)

    8. \(-6 \sec \left(\theta\right)+4=-8\)

    9. \(3 \cot \left(\theta\right)+\frac{2}{\sqrt{3}}=-\frac{1}{\sqrt{3}}\)

    10. \(0=4\sqrt{2}+8 \sin \left(\theta\right)\)

    11. \(\sqrt{3} \cos \left(\theta\right)=-\frac{3}{2}\)

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

    13. \(1-\cos ^2 \left(\theta\right)=0\)

    14. \(4 \sec ^2 \left(\theta\right)-3=0\)

    15. \(0=1.5-0.75 \csc ^2 \left(\theta\right)\)

    16. \(1-\cot ^2 \left(\theta\right)=0\)

    17. \(0=-6 + 18 \tan ^2 \left(\theta\right)\)

    18. \(\tan^5\left( x \right)=\tan\left( x \right)\)

    19. \(2 \sin^2 \left(\theta\right)+3\sin \left(\theta\right)=-1\)

    20. \( 3 = 6 \cos^2 \left(\theta\right) -3\cos \left(\theta\right)\)

    21. \(8\sin^2\left( x \right)+6\sin\left( x \right)+1=0\)

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

    23. \( 2\cos^2\left( \theta \right) +7\cos\left( \theta \right) + 3 = 0 \)

    24. \( -2 = \csc^2\left( \theta \right) - 3\csc\left( \theta \right)\)

    25. \( \sec^2\left( \theta \right) - 2 = - \sec\left( \theta \right) \)

    26. \(\sqrt{3} \tan \left(2 \alpha\right)=3\)

    27. \(2 \sin \left(2 \beta\right)=1\)

    28. \(-6 \cos \left(2 \theta\right)=6\)

    29. \(3 \tan \left(3 w\right)=\sqrt{3}\)

    30. \(2 \tan \left(3 u\right)=-2\)

    31. \(\frac{5}{7} \tan \left(\pi x\right)+11=11\)

    32. \(8 \sin \left(2 \theta-\frac{\pi}{6}\right)=-4\)

    33. \(6 \cos \left(3 \theta-\frac{\pi}{2}\right)=-3 \sqrt{2}\)

    34. \(3 \tan \left(\frac{w}{2}+\frac{\pi}{4}\right)+4=1\)

    35. \(50 \sec \left(\frac{t}{\pi}+6\right)-10=-110\)

    36. \(2 \tan \left(2 \pi \beta\right)+5=3\)

    37. \(160 \sin (\pi \phi-1)+10=90\)

    38. \(2 \cos \left(3 t+\frac{\pi}{4}\right)=\sqrt{3}\)

    39. \(2-\tan \left(2 x-\frac{\pi}{3}\right)=2\)

    40. \(7 \csc \left(\frac{\phi}{2}+\frac{3 \pi}{4}\right)+3=-4\)

    For the following exercises, solve the equation for \(0^{\circ} \leq \theta \leq 360^{\circ}\). Round your answers to two decimal places.

    1. \(4=6 \sin \left(\theta\right)\)

    2. \(3=5 \cos \left(\theta\right)\)

    3. \(\frac{1}{2} \tan \left(\theta\right)-1=-3\)

    4. \(7 \csc \left(\theta\right)+2=-16\)

    5. \(3 \sec \left(\theta\right)-2=5\)

    6. \(2=5-\frac{1}{3} \cot \left(\theta\right)\)

    7. \(6 \cos ^2 \left(\theta\right)=2\)

    8. \(2-7 \sin ^2 \left(\phi\right)=1\)

    9. \( -4\tan^2\left( \theta \right) + 5 = -4 \)

    10. \( 8 - 2\sec^2\left( \theta \right) = -2 \)

    11. \( 3 = 9 - 2 \csc^2\left( \theta \right) \)

    12. \( 9\cot^2\left( \theta \right) - 9 = -5 \)

    13. \(5 \sin ^2 \left(\theta\right)+\sin \left(\theta\right)=0\)

    14. \(10 \cos ^2 \left(\theta\right)-7 \cos \left(\theta\right)+1=0\)

    15. \(4 \tan ^2 \left(\theta\right)=\tan \left(\theta\right)\)

    16. \(\cot ^2 \left(\theta\right)-2 \cot \left(\theta\right)=15\)

    17. \(5 \sin \left(3 \theta\right)-3=-4\)

    18. \(9 \cos \left(2 \theta\right)+1=6\)

    19. \(8 \tan \left(4 t\right)+1=-11\)

    20. \(6 \csc \left(2 r\right)+2=13\)

    For the following exercises, solve the equation for \(0 \leq \theta \leq 2\pi\). Round your answers to two decimal places.

    1. \( 5\sin\left( \theta \right) - 4 = -6 \)

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

    3. \( -2\tan\left( \theta \right) + 5 = 6 \)

    4. \( -4 = 5 - \csc\left( \theta \right) \)

    5. \( 3\sec\left( \theta \right) - 2 = -10 \)

    6. \( \frac{1}{3}\cot\left( \theta \right) + 1 = -2 \)

    7. \(5 \sin ^2 \left(\theta\right)=2\)

    8. \(3-6 \cos ^2 \left(\phi\right)=-1\)

    9. \( 4-4\tan^2\left( \theta \right) = -5 \)

    10. \( 4 - \sec^2\left( \theta \right) = -1 \)

    11. \( -1 + 2 \csc^2\left( \theta \right) = 5\)

    12. \( \frac{9}{2}\cot^2\left( \theta \right) - 1 = 1 \)

    13. \(\sin ^2 \left(\theta\right)+8 \sin \left(\theta\right)+7=0\)

    14. \(\cos ^2 \left(\theta\right)-4 \cos \left(\theta\right)+3=0\)

    15. \(\tan ^2 \left(\theta\right)-5 \tan \left(\theta\right)+6=0\)

    16. \(6 \cot ^2 \left(\theta\right)-\cot \left(\theta\right)-1=0\)

    17. \(150 \sin \left(3 s\right)=27\)

    18. \(7 \cos \left(2 t\right)-3=2\)

    19. \(3=3 \tan \left(4 x\right)+4\)

    20. \(2-8 \sec \left(3 t\right)=-10\)

    21. \(9 \sin \left(\frac{\pi}{2}(t-1)\right)+5=-1\)

    22. \(3-5 \cos (2 \phi-1)=6\)

    23. \(14 \tan (\pi \beta-4)+31=10\)

    24. \(0.3 \csc \left(\frac{\pi}{2}(\alpha+0.1)\right)+0.4=1.9\)

    25. \(2 \sec \left(\frac{\pi}{4}\left(t+\frac{1}{4}\right)\right)+3=-7\)

    26. \(6 \cot \left(\frac{\pi}{3}(\theta-1)\right)+4=5\)

    27. \(120 \sin \left(\frac{\pi}{3}(t-0.2)\right)+21=-3\)

    28. \(16 \cos (3 t-1)+4=-8\)

    29. \(23-24 \tan (\pi x+2)=17\)

    30. \(3 \csc \left(3 w-\frac{\pi}{3}\right)+1=6\)

    31. \(12 \sec \left(\frac{\pi}{2}(t+0.3)\right)-7=9\)

    32. \(8 \cot \left(4 t-\frac{\pi}{3}\right)-24=1\)

    Proofs

    1. What is the angle in the first quadrant with reference angle \( \theta \)? What is the angle in the third quadrant with reference angle \(\theta\)? What is the difference of these angles? Explain how this fact shows that the solutions of \(\tan \left(\theta\right)=k\), for \(k>0\), differ by \(\pi\).

    2. What is the angle in the second quadrant with reference angle \(\theta\)? What is the angle in the fourth quadrant with reference angle \(\theta\)? What is the difference of these angles? Explain how this fact shows that the solutions \(\tan \left(\theta\right)=k\), for \(k<0\), differ by \(\pi\).

    Applications

    1. A sinusoidal function can approximate the population of deer in Marquette County over a typical year. The population reached a maximum of 50,000 deer on September 1st and a minimum of 42,000 on March 1st.

      1. Write a formula for the function \(P(t)\) that gives the deer population on the first of each month, where \(t=0\) is September 1.

      2. When is the deer population 45,000? Give exact expressions and approximations rounded to two decimal places.

      3. Graph your function over one period, and label the points that correspond to a deer population of 45,000. Is the population greater or less than 45,000 between the two solutions?

    2. The percent of the moon visible from Earth is a sinusoidal function ranging from \(0 \%\) to \(100 \%\), with a period of 29.5 days.

      1. Write a formula for the function \(f(t)\) that gives the percent of the moon that is visible, if a new moon (\(0 \%\) visible) occurs at \(t=0\) days.

      2. When is \(25 \%\) of the moon visible? Give approximations rounded to two decimal places.

      3. Graph your function over one period, and label the points corresponding to a quarter moon. Is more or less than \(25 \%\) of the moon visible between the two solutions you found in part (b)?

    3. A Ferris wheel has a diameter of 20 meters and completes one revolution every 60 seconds. Jorge is at the lowest position of the Ferris wheel, 1 meter above the ground, at \(t=0\) seconds.

      1. Write a formula for the function \(h(t)\) that gives Jorge's altitude in meters after \(t\) seconds.

      2. When is Jorge at an altitude of 18 meters during his first revolution? Give exact expressions and approximations rounded to two decimal places.

      3. Graph your function over one period, labeling the points that correspond to an altitude of 18 meters. Is Jorge above or below 18 meters between the two solutions you found in part (b)?

    4. High tides occur every 12.2 hours at Point Lookout. The depth of the water at the end of Roy’s dock is 2.6 meters at high tide and 1.8 meters at low tide.

      1. Write a formula for the function \(d(t)\) that gives the depth of the water \(t\) hours after last night’s high tide.

      2. When is the water at the end of the dock 2 meters deep? Give exact approximations rounded to two decimal places.

      3. Graph your function over one period, labeling the points that correspond to a depth of 2 meters. Is the water depth greater or less than 2 meters between the two solutions you found in part (b)?

    This page titled 10.3.2: Homework was last modified on Tue, 08 Jul 2025 17:46:25 GMT and is shared under a not declared license and was authored, remixed, and/or curated by Roy Simpson.