[Video] Solving Trig Equations

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In this series of videos, we will be aimed at learning how solving trigonometric equations.

The following videos will cover the solving trigonometric equations:

Introduction to Trigonometric Equations: What is a trigonometric equation and how to use the unit circle to solve trigonometric equations

(Click the arrow to the right to see the Introduction video.)

Question 1: Solving basic trigonometric equations

(Click the arrow to the right to see the first question.)

(Click the arrow to the right to see the question posed at the end of the video.)

What are all solutions to $\sec {x} - \sqrt {2} = 0$ ?

$\frac {\pi }{4} + 2\pi n$ and $\frac {7\pi }{4} + 2\pi n$ for all integers $n$ $\frac {\pi }{4} + \pi n$ and $\frac {7\pi }{4} + \pi n$ for all integers $n$ $\frac {\pi }{4}$ and $\frac {7\pi }{4}$ $\frac {4}{\pi } + 2\pi n$ and $\frac {4}{7\pi } + 2\pi n$ for all integers $n$ $\frac {4}{\pi } + \pi n$ and $\frac {4}{7\pi } + \pi n$ for all integers $n$ $\frac {4}{\pi }$ and $\frac {4}{7\pi }$ $\frac {\pi }{4} + 2\pi n$ and $\frac {3\pi }{4} + 2\pi n$ for all integers $n$ $\frac {\pi }{4} + \pi n$ and $\frac {3\pi }{4} + \pi n$ for all integers $n$ $\frac {\pi }{4}$ and $\frac {3\pi }{4}$ $\frac {3\pi }{4} + 2\pi n$ and $\frac {5\pi }{4} + 2\pi n$ for all integers $n$ $\frac {3\pi }{4} + \pi n$ and $\frac {5\pi }{4} + \pi n$ for all integers $n$ $\frac {3\pi }{4}$ and $\frac {5\pi }{4}$ $\frac {5\pi }{4} + 2\pi n$ and $\frac {7\pi }{4} + 2\pi n$ for all integers $n$ $\frac {5\pi }{4} + \pi n$ and $\frac {7\pi }{4} + \pi n$ for all integers $n$ $\frac {5\pi }{4}$ and $\frac {7\pi }{4}$ There is no way to find all of the solutions. None of the above.

(Click the arrow to the right to see an example.)

Example 1

Question 2: Solving trigonometric equations

(Click the arrow to the right to see the second question.)

(Click the arrow to the right to see the question posed at the end of the video.)

What are all solutions to $\sin (\theta ) + \cos (\theta )=0$ ?

$\frac {\pi }{4} + 2\pi n$ and $\frac {5\pi }{4} + 2\pi n$ for all integers $n$ $\frac {\pi }{4} + \pi n$ for all integers $n$ $\frac {\pi }{4}$ and $\frac {3\pi }{4}$ $\frac {\pi }{4} + 2\pi n$ and $\frac {3\pi }{4} + 2\pi n$ for all integers $n$ $\frac {\pi }{4} + \pi n$ and $\frac {3\pi }{4} + \pi n$ for all integers $n$ $\frac {\pi }{4}$ and $\frac {5\pi }{4}$ $\frac {3\pi }{4} + 2\pi n$ and $\frac {7\pi }{4} + 2\pi n$ for all integers $n$ $\frac {3\pi }{4} + \pi n$ and $\frac {7\pi }{4} + \pi n$ for all integers $n$ $\frac {3\pi }{4}$ and $\frac {7\pi }{4}$ $\frac {5\pi }{4} + 2\pi n$ and $\frac {7\pi }{4} + 2\pi n$ for all integers $n$ $\frac {5\pi }{4} + \pi n$ and $\frac {7\pi }{4} + \pi n$ for all integers $n$ $\frac {5\pi }{4}$ and $\frac {7\pi }{4}$ There is no way to find all of the solutions. None of the above.

Question 3: Solving more difficult trigonometric equations

(Click the arrow to the right to see the third question.)

(Click the arrow to the right to see the question posed at the end of the video.)

What are all solutions to $6\cos ^2(x) - 3 = 0$ ? (Select all correct answers, if applicable.)

$0 + 2\pi n$ for integers $\frac {\pi }{6} + 2\pi n$ for all integers $n$ $\frac {\pi }{4} + 2\pi n$ for all integers $n$ $\frac {\pi }{3} + 2\pi n$ for all integers $n$ $\frac {\pi }{2} + 2\pi n$ for all integers $n$ $\frac {2\pi }{3} + 2\pi n$ for all integers $n$ $\frac {3\pi }{4} + 2\pi n$ for all integers $n$ $\frac {5\pi }{6} + 2\pi n$ for all integers $n$ $\pi + 2\pi n$ for all integers $n$ $\frac {7\pi }{6} + 2\pi n$ for all integers $n$ $\frac {5\pi }{4} + 2\pi n$ for all integers $n$ $\frac {4\pi }{3} + 2\pi n$ for all integers $n$ $\frac {3\pi }{2} + 2\pi n$ for all integers $n$ $\frac {5\pi }{3} + 2\pi n$ for all integers $n$ $\frac {7\pi }{4} + 2\pi n$ for all integers $n$ $\frac {11\pi }{6} + 2\pi n$ for all integers $n$ There is no way to find all of the solutions. There are no solutions. None of the above.

(Click the arrow to the right to see an example.)

Example 3

Question 4: Solving more difficult trigonometric equations

(Click the arrow to the right to see the fourth question.)

(Click the arrow to the right to see the question posed at the end of the video.)

What are all solutions to $\tan (\alpha )\cos (\alpha ) = \tan (\alpha )$ from $0\leq \alpha <2\pi$ ?
(Select all correct answers, if applicable.)
(Hint: If you divide by $\tan (\alpha )$ you may lose solutions because when you do this you are assuming $\tan (\alpha ) \neq 0$ .)

$0$ $\frac {\pi }{6}$ $\frac {\pi }{4}$ $\frac {\pi }{3}$ $\frac {\pi }{2}$ $\frac {2\pi }{3}$ $\frac {3\pi }{4}$ $\frac {5\pi }{6}$ $\pi$ $\frac {7\pi }{6}$ $\frac {5\pi }{4}$ $\frac {4\pi }{3}$ $\frac {3\pi }{2}$ $\frac {5\pi }{3}$ $\frac {7\pi }{4}$ $\frac {11\pi }{6}$ There is no way to find all of the solutions. There are no solutions. None of the above.

(Click the arrow to the right to see an example.)

Example 4

Question 5: Solving even more difficult trigonometric equations

(Click the arrow to the right to see the fifth question.)

(Click the arrow to the right to see the question posed at the end of the video.)

What are all solutions to $-2\cos ^2(\theta ) - \cos (\theta ) = -1$ ? (Select all correct answers, if applicable.)

$0 + 2\pi n$ for all integers $n$ $\frac {\pi }{6} + 2\pi n$ for all integers $n$ $\frac {\pi }{4} + 2\pi n$ for all integers $n$ $\frac {\pi }{3} + 2\pi n$ for all integers $n$ $\frac {\pi }{2} + 2\pi n$ for all integers $n$ $\frac {2\pi }{3} + 2\pi n$ for all integers $n$ $\frac {3\pi }{4} + 2\pi n$ for all integers $n$ $\frac {5\pi }{6} + 2\pi n$ for all integers $n$ $\pi + 2\pi n$ for all integers $n$ $\frac {7\pi }{6} + 2\pi n$ for all integers $n$ $\frac {5\pi }{4} + 2\pi n$ for all integers $n$ $\frac {4\pi }{3} + 2\pi n$ for all integers $n$ $\frac {3\pi }{2} + 2\pi n$ for all integers $n$ $\frac {5\pi }{3} + 2\pi n$ for all integers $n$ $\frac {7\pi }{4} + 2\pi n$ for all integers $n$ $\frac {11\pi }{6} + 2\pi n$ for all integers $n$ There is no way to find all of the solutions. There are no solutions. None of the above.

(Click the arrow to the right to see an example.)

Example 5

Question 6: Solving trigonometric equations by rewriting trigonometric functions

(Click the arrow to the right to see the sixth question.)

(Click the arrow to the right to see the question posed at the end of the video.)

What are all solutions to $2\sin (x) - 2\csc (x) = 0$ ? (Select all correct answers, if applicable.)

(Click the arrow to the right to see an example.)

Example 6

Question 7: Solving trigonometric equations with non-standard arguments (or inputs)

(Click the arrow to the right to see the seventh question.)

(Click the arrow to the right to see the question posed at the end of the video.)

What are all solutions to $2\sin (3x)=0$ ? (Select all correct answers, if applicable.)

(Click the arrow to the right to see an example.)

Example 7