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To solve for the values of $\theta$ where $\cot(\theta) = 1$ on the unit circle for

txt1

txt1

txt1

\leq \theta < 2\pi$, we start by recalling that $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$. Hence, $\cot(\theta) = 1$ implies $\frac{\cos(\theta)}{\sin(\theta)} = 1$, or $\cos(\theta) = \sin(\theta)$.

On the unit circle, the equation $\cos(\theta) = \sin(\theta)$ holds when $\theta = \frac{\pi}{4} + k\pi$ for integer $k$. We need the values of $\theta$ in the interval

txt1

txt1

txt1

\leq \theta < 2\pi$. Thus, the possible values of $\theta$ are $\frac{\pi}{4}$ and $\frac{5\pi}{4}$.

Therefore, the values of $\theta$ where $\cot(\theta) = 1$ on the unit circle for

txt1

txt1

txt1

\leq \theta < 2\pi$ are:

$\theta = \frac{\pi}{4}, \frac{5\pi}{4}$

To determine the angles $ heta$ for which $cot( heta) = 1$ on the unit circle within the interval

txt2

txt2

txt2

leq heta < 2pi$, we utilize the identity $cot( heta) = frac{cos( heta)}{sin( heta)}$. Setting $frac{cos( heta)}{sin( heta)} = 1$ results in $cos( heta) = sin( heta)$.

The equation $cos( heta) = sin( heta)$ occurs at angles of $ heta = frac{pi}{4} + kpi$ for integer $k$. Considering

txt2

txt2

txt2

leq heta < 2pi$, we identify the specific solutions $ heta = frac{pi}{4}$ and $ heta = frac{5pi}{4}$ as valid.

Hence, the solutions to the given equation within the specified interval are:

$ heta = frac{pi}{4}, frac{5pi}{4}$

Given $cot( heta) = 1$ on the unit circle and

txt3

txt3

txt3

leq heta < 2pi$, we start with $cot( heta) = frac{cos( heta)}{sin( heta)} = 1$, implying $cos( heta) = sin( heta)$.

This relationship is true for $ heta = frac{pi}{4} + kpi$. Within

txt3

txt3

txt3

leq heta < 2pi$, the angles are $frac{pi}{4}$ and $frac{5pi}{4}$.

Thus, the values are:

$ heta = frac{pi}{4}, frac{5pi}{4}$

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