Find all solutions of the equation tan(θ) = -2/3 in the interval [0, 2π]. If there is more than one solution, separate them with commas.
Understand the Problem
The question is asking for all the solutions to the equation involving the tangent function, specifically finding angles θ in the interval from 0 to 2π where tan(θ) equals -2/3. It requires solving a trigonometric equation and identifying possible angles.
Answer
The values of $\theta$ are approximately $2.553$ and $5.695$ radians.
Answer for screen readers
The solutions for the equation $\tan(\theta) = -\frac{2}{3}$ in the interval $[0, 2\pi]$ are approximately: $$ \theta \approx 2.553, 5.695 $$
Steps to Solve
- Identify the reference angle
To find the angles where $\tan(\theta) = -\frac{2}{3}$, first, determine the reference angle $\alpha$ corresponding to the positive ratio of $\frac{2}{3}$.
Using the arctangent function: $$ \alpha = \tan^{-1}\left(\frac{2}{3}\right) $$
- Determine the quadrants
Since the tangent function is negative, the solutions will be in the second and fourth quadrants.
- Find the angles in the specific quadrants
For the second quadrant: $$ \theta_1 = \pi - \alpha $$ For the fourth quadrant: $$ \theta_2 = 2\pi - \alpha $$
- Calculate the numerical values
Using a calculator to find $\alpha$: $$ \alpha \approx \tan^{-1}\left(\frac{2}{3}\right) \approx 0.588 \text{ radians} $$
Now compute: $$ \theta_1 \approx \pi - 0.588 \approx 2.553 \text{ radians} $$ $$ \theta_2 \approx 2\pi - 0.588 \approx 5.695 \text{ radians} $$
- List all solutions
The solutions in the interval $[0, 2\pi]$ are: $$ \theta \approx 2.553, 5.695 $$
The solutions for the equation $\tan(\theta) = -\frac{2}{3}$ in the interval $[0, 2\pi]$ are approximately: $$ \theta \approx 2.553, 5.695 $$
More Information
The tangent function's periodic behavior and the corresponding angle calculations are essential in trigonometric equations. This illustrates how angle solutions can be derived from reference angles and their positions on the unit circle, particularly when working with negative values.
Tips
- Not identifying the correct quadrants: Always check where the tangent is negative.
- Forgetting to convert angles: Ensure angles are in radians if requested, particularly when using a calculator.
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