Given the unit circle, find the angles $ heta$ between Define the unit circle in trigonometry$ and $2pi$ for which the secant function $sec( heta)$ equals 2, and provide a step-by-step explanation for your answer.

We start by recalling the definition of the secant function: $\sec(\theta) = \frac{1}{\cos(\theta)}$. Therefore, the given condition $\sec(\theta) = 2$ translates to:

$\frac{1}{\cos(\theta)} = 2$

Solving for $\cos(\theta)$, we get:

$\cos(\theta) = \frac{1}{2}$

Now, we need to find the angles $\theta$ in the interval $[0, 2\pi)$ such that $\cos(\theta) = \frac{1}{2}$. These angles can be found using the unit circle:

$\theta = \frac{\pi}{3} + 2k\pi \quad \text{and} \quad \theta = \frac{5\pi}{3} + 2k\pi \quad \text{for integers k}$

Considering the interval

txt1

txt1

txt1

\leq \theta < 2\pi$, we have:

$\theta = \frac{\pi}{3} \quad \text{and} \quad \theta = \frac{5\pi}{3}$

Therefore, the angles $\theta$ are $\frac{\pi}{3}$ and $\frac{5\pi}{3}$.

To solve the problem, we start from the definition of the secant function:

$sec( heta) = frac{1}{cos( heta)} = 2$

This implies:

$cos( heta) = frac{1}{2}$

Next, we determine the angles $ heta$ that satisfy $cos( heta) = frac{1}{2}$ within the interval $[0, 2pi)$.

We know from the unit circle that:

$cos( heta) = frac{1}{2}$

at angles $ heta = frac{pi}{3}$ and $ heta = frac{5pi}{3}$.

Hence, the solutions to the given problem are:

$ heta = frac{pi}{3} quad ext{and} quad heta = frac{5pi}{3}$

To find $ heta$ such that $sec( heta) = 2$, we start with:

$sec( heta) = frac{1}{cos( heta)} = 2$

This gives:

$cos( heta) = frac{1}{2}$

From the unit circle:

$ heta = frac{pi}{3}, frac{5pi}{3}$

Start Using PopAi Today