Question
Solve the equation on the interval \( 0 \leq 0<2 \pi \) \[ (\tan 0-1)(\cos \theta+1)=0 \] Select the correct choice below and rill in any answer boxes in your choice. A. The solution set is (Simplify your answer. Type an oxact answer, using \( \pi \) as needed. Type your answer in radians. Use integers or B. There is no solution on this interval.
Ask by Fleming Little. in the United States
Dec 03,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solution set is \( \theta = \pi + 2k\pi \), where \( k \) is an integer.
Solution
Solve the equation \( (\tan 0-1)(\cos \theta+1)=0 \).
Solve the equation by following steps:
- step0: Solve for \(\theta\):
\(\left(\tan\left(0\right)-1\right)\left(\cos\left(\theta \right)+1\right)=0\)
- step1: Simplify:
\(-\cos\left(\theta \right)-1=0\)
- step2: Move the constant to the right side:
\(-\cos\left(\theta \right)=0+1\)
- step3: Remove 0:
\(-\cos\left(\theta \right)=1\)
- step4: Multiply both sides:
\(-\cos\left(\theta \right)\left(-1\right)=1\times \left(-1\right)\)
- step5: Calculate:
\(\cos\left(\theta \right)=1\times \left(-1\right)\)
- step6: Calculate:
\(\cos\left(\theta \right)=-1\)
- step7: Use the inverse trigonometric function:
\(\theta =\arccos\left(-1\right)\)
- step8: Calculate:
\(\theta =\pi \)
- step9: Add the period:
\(\theta =\pi +2k\pi ,k \in \mathbb{Z}\)
The solution to the equation \( (\tan 0-1)(\cos \theta+1)=0 \) on the interval \( 0 \leq \theta < 2\pi \) is \( \theta = \pi + 2k\pi \), where \( k \) is an integer.
Therefore, the correct choice is A. The solution set is \( \theta = \pi + 2k\pi \), where \( k \) is an integer.
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Simplify this solution Bonus Knowledge
To solve the equation \( (\tan \theta - 1)(\cos \theta + 1) = 0 \), we set each factor to zero: 1. \( \tan \theta - 1 = 0 \) implies \( \tan \theta = 1 \). The solutions to this in the interval \( 0 \leq \theta < 2\pi \) are \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{5\pi}{4} \). 2. \( \cos \theta + 1 = 0 \) implies \( \cos \theta = -1 \). The solution to this in the same interval is \( \theta = \pi \). Combining both results, the solution set is \( \left\{ \frac{\pi}{4}, \pi, \frac{5\pi}{4} \right\} \). Thus, the correct answer is: A. The solution set is \( \left\{ \frac{\pi}{4}, \pi, \frac{5\pi}{4} \right\} \).
