Find all solutions of the equation in the interval [0, 2π], csc θ = 5/3. If there is more than one solution, separate them with commas.
Understand the Problem
The question is asking to find all angles θ in the interval [0, 2π] where the cosecant of θ equals 5/3. This involves solving the equation and expressing the solutions in the specified interval.
Answer
The solutions are approximately \( 0.6435, 2.4981 \).
Answer for screen readers
The solutions are approximately: $$ \theta \approx 0.6435, 2.4981 $$
Steps to Solve
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Rewrite the cosecant function The cosecant function is the reciprocal of the sine function. Thus, we can rewrite the equation: $$ \csc \theta = \frac{5}{3} $$ as $$ \sin \theta = \frac{3}{5} $$
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Find the reference angle To find the angle whose sine is ( \frac{3}{5} ), we can use the inverse sine function: $$ \theta_{\text{ref}} = \sin^{-1} \left( \frac{3}{5} \right) $$
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Determine the angles in the specified interval Since sine is positive in the first and second quadrants, we find:
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The first angle: $$ \theta_1 = \theta_{\text{ref}} $$
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The second angle: $$ \theta_2 = \pi - \theta_{\text{ref}} $$
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Evaluate the reference angle To find the numerical values of the angles, compute ( \theta_{\text{ref}} ): If we assume ( \theta_{\text{ref}} \approx 0.6435 ) radians, then: $$ \theta_1 \approx 0.6435 $$ $$ \theta_2 \approx \pi - 0.6435 \approx 2.4981 $$
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Final solutions The final solutions in the interval ( [0, 2\pi] ) are: $$ \theta \approx 0.6435, 2.4981 $$
The solutions are approximately: $$ \theta \approx 0.6435, 2.4981 $$
More Information
The sine function has positive values in the first and second quadrants, which is why we found two solutions: one in each quadrant. Calculators or tables may provide the reference angle directly.
Tips
- Confusing the cosecant function with other trigonometric functions. Remember that ( \csc \theta = \frac{1}{\sin \theta} ).
- Not considering both quadrants for sine values since sine is positive in two quadrants.
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