”Given

We start by setting the equation:

$\sin(\theta) = \cos(\theta)$

We know from trigonometric identities that:

$\sin(\theta) = \cos(\theta)$

Dividing both sides by $\cos(\theta)$ (assuming $\cos(\theta) \neq 0$):

$\frac{\sin(\theta)}{\cos(\theta)} = 1$

Using the identity $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$, we get:

$\tan(\theta) = 1$

This implies that:

$\theta = \frac{\pi}{4} + k\pi \text{ for some integer } k$

Since we need $\theta$ in the interval $[0, 2\pi)$, we find:

$\theta = \frac{\pi}{4} \text{ or } \theta = \frac{5\pi}{4}$

Therefore, the angles $\theta$ are:

$\theta = \frac{\pi}{4} \text{ and } \theta = \frac{5\pi}{4}$

To solve $sin( heta) = cos( heta)$, we consider the unit circle where $sin( heta)$ and $cos( heta)$ are equal.

As these functions are equal when $ heta = frac{pi}{4}$ and $ heta = frac{5pi}{4}$, we can solve for $ heta$ by examining:

$sin( heta) – cos( heta) = 0$

This leads us to:

$sin( heta) = cos( heta)$

Or equivalently:

$ an( heta) = 1$

Thus:

$ heta = frac{pi}{4} + npi , ext{for}, n in mathbb{Z}$

Within the interval $[0, 2pi)$, the valid angles are:

$ heta = frac{pi}{4} ext{ and } heta = frac{5pi}{4}$

Given $sin( heta) = cos( heta)$, we divide both sides by $cos( heta)$:

$frac{sin( heta)}{cos( heta)} = 1$

Therefore, $ an( heta) = 1$, leading to:

$ heta = frac{pi}{4} + kpi$

Within $[0, 2pi)$, we have:

$ heta = frac{pi}{4} ext{ and } heta = frac{5pi}{4}$

Start Using PopAi Today