Find all solutions of the equation in the interval $[0,2 \pi)$.
$\csc \theta-2=0$
Write your answer in radians in terms of $\pi$. If there is more than one solution, separate them with commas.
$\square$
$\theta=$
$\square$
Answer (2)
Rewrite the equation csc θ − 2 = 0 as sin θ = 2 1 .
Find the reference angle: 6 π .
Determine the solutions in the interval [ 0 , 2 π ) : 6 π and 6 5 π .
The solutions are 6 π , 6 5 π .
Explanation
Problem Analysis We are given the equation csc θ − 2 = 0 and we need to find all solutions for θ in the interval [ 0 , 2 π ) .
Rewriting the Equation First, let's rewrite the equation in terms of sine. Since csc θ = s i n θ 1 , the equation becomes: sin θ 1 − 2 = 0 Adding 2 to both sides, we get: sin θ 1 = 2 Taking the reciprocal of both sides, we have: sin θ = 2 1
Finding the Reference Angle Now we need to find all angles θ in the interval [ 0 , 2 π ) such that sin θ = 2 1 . We know that sin θ is positive in the first and second quadrants. The reference angle for θ is the angle whose sine is 2 1 , which is 6 π .
Finding Solutions in the Interval In the first quadrant, the angle is simply the reference angle: θ 1 = 6 π In the second quadrant, the angle is: θ 2 = π − 6 π = 6 6 π − 6 π = 6 5 π
Final Solutions Therefore, the solutions in the interval [ 0 , 2 π ) are 6 π and 6 5 π .
Examples
Understanding trigonometric equations like csc θ − 2 = 0 is crucial in fields like physics and engineering. For instance, when analyzing the motion of a pendulum or the behavior of alternating current in an electrical circuit, you often encounter equations involving trigonometric functions. Solving these equations allows you to determine key parameters such as the angle of the pendulum at a specific time or the phase difference between voltage and current in the circuit. This ensures accurate predictions and efficient designs in various real-world applications.
The solutions to the equation csc θ − 2 = 0 in the interval [ 0 , 2 π ) are 6 π and 6 5 π . These solutions occur where sin θ = 2 1 .
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