Find all solutions of the equation in the interval $[0,2 \pi)$.
$\cot \theta-1=0$
Write your answer in radians in terms of $\pi$.
If there is more than one solution, separate them with commas.
$\theta=$
Answer (2)
Rewrite the given equation cot θ − 1 = 0 as cot θ = 1 .
Recognize that cot θ = s i n θ c o s θ , so we need to find angles where cos θ = sin θ .
Identify the angles in the interval [ 0 , 2 π ) where cos θ = sin θ , which are 4 π and 4 5 π .
The solutions are 4 π , 4 5 π .
Explanation
Problem Analysis We are given the equation cot θ − 1 = 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 as cot θ = 1 . Recall that cot θ = s i n θ c o s θ . Therefore, we are looking for angles θ where cos θ = sin θ .
Finding the Angles We need to find the angles θ in the interval [ 0 , 2 π ) such that cos θ = sin θ . This occurs when θ = 4 π and θ = 4 5 π . To see this, consider the unit circle. The coordinates of a point on the unit circle are ( cos θ , sin θ ) . We want to find the angles where the x and y coordinates are equal. In the first quadrant, this occurs at 4 π , where cos 4 π = sin 4 π = 2 2 . In the third quadrant, this occurs at 4 5 π , where cos 4 5 π = sin 4 5 π = − 2 2 .
Solutions Therefore, the solutions to the equation cot θ − 1 = 0 in the interval [ 0 , 2 π ) are θ = 4 π and θ = 4 5 π .
Examples
Understanding trigonometric equations like cot θ − 1 = 0 is crucial in various fields such as physics and engineering. For instance, when analyzing the motion of a pendulum, the angle θ it makes with the vertical can be modeled using trigonometric functions. Solving equations involving these functions helps determine specific angles at certain times, which is essential for predicting the pendulum's behavior. Similarly, in electrical engineering, alternating current (AC) circuits involve sinusoidal functions, and solving trigonometric equations helps determine phase angles and current values at different points in time.
The solutions to the equation cot θ − 1 = 0 in the interval [ 0 , 2 π ) are 4 π and 4 5 π .
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