Find all solutions of the equation in the interval [0, 2π]. csc(θ) = -4/3. If there is more than one solution, separate them with commas.
Understand the Problem
The question is asking to find all the angles θ in the specified interval [0, 2π] that satisfy the equation csc(θ) = -4/3. This involves solving for θ based on the properties of the cosecant function and its relationship to the sine function.
Answer
The angles are approximately $3.9897$, $5.4351$.
Answer for screen readers
The solutions are:
$$ \theta \approx 3.9897, 5.4351 $$
Steps to Solve
-
Rewrite Cosecant in Terms of Sine
To solve the equation ( \csc(\theta) = -\frac{4}{3} ), we first rewrite it in terms of sine. Recall that ( \csc(\theta) = \frac{1}{\sin(\theta)} ). Therefore, the equation becomes:
$$ \sin(\theta) = -\frac{3}{4} $$ -
Determine the Quadrants for the Solution
Since ( \sin(\theta) ) is negative, we know that ( \theta ) must be in the third or fourth quadrants. -
Find the Reference Angle
To find the reference angle ( \theta_r ), take the inverse sine of the positive value of ( -\frac{3}{4} ):
$$ \theta_r = \arcsin\left(\frac{3}{4}\right) $$
Calculating this gives approximately ( \theta_r \approx 0.8481 ) radians. -
Calculate the Angles in the Specified Interval
Using the reference angle and the known quadrants:
-
For the third quadrant:
$$ \theta_1 = \pi + \theta_r \approx \pi + 0.8481 \approx 3.9897 $$ -
For the fourth quadrant:
$$ \theta_2 = 2\pi - \theta_r \approx 2\pi - 0.8481 \approx 5.4351 $$
-
Final Solutions
Thus, the solutions in the interval ([0, 2\pi]) are:
$$ \theta \approx 3.9897, 5.4351 $$
The solutions are:
$$ \theta \approx 3.9897, 5.4351 $$
More Information
The cosecant function is the reciprocal of the sine function. Hence, while solving, it's crucial to remember the signs of sine in different quadrants, which affects the solutions. The values ( 3.9897 ) and ( 5.4351 ) radians correspond to angles in the third and fourth quadrants, where sine is negative.
Tips
- Mixing up the signs of sine: Always check the quadrant you are in when determining the sign of sine.
- Failing to convert the reference angle back to the specific angle in the correct quadrant.
AI-generated content may contain errors. Please verify critical information
