Unit Circle

Calculate the value of tan(θ) for θ = 7π/4 using the unit circle

To find the value of $\tan(\theta)$ for $\theta = \frac{7\pi}{4}$ using the unit circle, we first need to determine the coordinates of the point on the unit circle corresponding to $\theta = \frac{7\pi}{4}$.

$\theta = \frac{7\pi}{4}$ corresponds to an angle of $315^\circ$ in standard position.

In the unit circle, this point is $\left( \cos\left(\frac{7\pi}{4}\right), \sin\left(\frac{7\pi}{4}\right) \right)$.

The coordinates at $\frac{7\pi}{4}$ are $( \frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} )$.

Therefore, $\tan\left(\frac{7\pi}{4}\right) $ can be calculated as:

$$ \tan\left(\frac{7\pi}{4}\right) = \frac{\sin\left(\frac{7\pi}{4}\right)}{\cos\left(\frac{7\pi}{4}\right)} = \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = -1 $$

Given a point on the unit circle, find its coordinates and the associated angle in radians, if the sine of the angle is equal to the cosine of the angle

Given $\sin(\theta) = \cos(\theta)$ for an angle $\theta$ on the unit circle:

We know that for an angle $\theta$ on the unit circle:

$$\sin^2(\theta) + \cos^2(\theta) = 1$$

Let $\sin(\theta) = \cos(\theta) = x$. Then,

$$x^2 + x^2 = 1$$

$$2x^2 = 1$$

$$x^2 = \frac{1}{2}$$

$$x = \pm \frac{1}{\sqrt{2}}$$

Therefore, $\sin(\theta) = \cos(\theta) = \pm \frac{1}{\sqrt{2}}$.

The coordinates are $(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$ and $(-\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}})$.

For $\frac{1}{\sqrt{2}}$, the angle is:

$$\theta = \frac{\pi}{4} + 2n\pi, \text{ for any integer } n$$

For $-\frac{1}{\sqrt{2}}$, the angle is:

$$\theta = \frac{5\pi}{4} + 2n\pi, \text{ for any integer } n$$

Find the coordinates of a point on the unit circle where the x-coordinate is 1/2

The equation of the unit circle is given by:

$$x^2 + y^2 = 1$$

We are given that the x-coordinate is $\frac{1}{2}$. Substituting $x = \frac{1}{2}$ into the equation:

$$\left(\frac{1}{2}\right)^2 + y^2 = 1$$

$$\frac{1}{4} + y^2 = 1$$

Subtract $\frac{1}{4}$ from both sides:

$$y^2 = 1 – \frac{1}{4}$$

$$y^2 = \frac{3}{4}$$

Taking the square root of both sides:

$$y = \pm \sqrt{\frac{3}{4}}$$

$$y = \pm \frac{\sqrt{3}}{2}$$

Thus, the coordinates are:

$$(\frac{1}{2}, \frac{\sqrt{3}}{2})$$ and $$(\frac{1}{2}, -\frac{\sqrt{3}}{2})$$

Determine the coordinates of a point on the unit circle for a given angle

To determine the coordinates of a point on the unit circle for a given angle $\theta$, we use the fact that the unit circle has a radius of 1 and the coordinates can be expressed as $(\cos(\theta), \sin(\theta))$.

Let’s find the coordinates for $\theta = \frac{\pi}{4}$.

The cosine and sine of $\frac{\pi}{4}$ are as follows:

$$\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$$

$$\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$$

Thus, the coordinates of the point are:

$$\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$$

Calculate sine, cosine, and tangent values at specific angles on the unit circle

Given the angle $ \theta = \frac{2\pi}{3} $ radians, calculate $ \sin(\theta) $, $ \cos(\theta) $, and $ \tan(\theta) $.

Solution:

First convert the angle to degrees to understand its position on the unit circle: $\theta = \frac{2\pi}{3} $ radians = $120^\circ$.

From the unit circle, for $120^\circ$:

$$\sin(120^\circ) = \sin(\frac{2\pi}{3}) = \frac{\sqrt{3}}{2} $$

$$\cos(120^\circ) = \cos(\frac{2\pi}{3}) = -\frac{1}{2} $$

$$\tan(120^\circ) = \tan(\frac{2\pi}{3}) = -\sqrt{3} $$

Find the value of cosecant for a complex angle on the unit circle

To find the value of $\csc(\theta + i \phi)$ on the unit circle, we first recall that $\csc(z) = \frac{1}{\sin(z)}$ and we utilize the definition of the sine function for complex arguments.

Given $z = \theta + i \phi$, we have:

$$\sin(z) = \sin(\theta + i \phi)$$

Using the identity for sine of a complex number, we get:

$$\sin(\theta + i \phi) = \sin(\theta) \cosh(\phi) + i \cos(\theta) \sinh(\phi)$$

Therefore,

$$\csc(\theta + i \phi) = \frac{1}{\sin(\theta + i \phi)} = \frac{1}{\sin(\theta) \cosh(\phi) + i \cos(\theta) \sinh(\phi)}$$

Hence, the final value of $\csc(\theta + i \phi)$ is:

$$\csc(\theta + i \phi) = \frac{\sin(\theta) \cosh(\phi) – i \cos(\theta) \sinh(\phi)}{\sin^2(\theta) \cosh^2(\phi) + \cos^2(\theta) \sinh^2(\phi)}$$

Calculate the sine and cosine values for the angle π/4 on the unit circle

To find the sine and cosine values for the angle $\frac{\pi}{4}$ on the unit circle, we use the fact that the unit circle has a radius of 1 and the coordinates of the point on the unit circle corresponding to this angle are $(\cos\theta, \sin\theta)$.

For $\theta = \frac{\pi}{4}$, the coordinates are:

$$ (\cos\frac{\pi}{4}, \sin\frac{\pi}{4}) $$

We know from trigonometric identities:

$$ \cos\frac{\pi}{4} = \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2} $$

Thus, the cosine and sine values for the angle $\frac{\pi}{4}$ are both $\frac{\sqrt{2}}{2}$.

Given a point on the unit circle at an angle θ = π/4, find the coordinates of the point

We know that the coordinates of a point on the unit circle are given by $(\cos(\theta), \sin(\theta))$.

Given $\theta = \frac{\pi}{4}$:

$$\cos\left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}$$

$$\sin\left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}$$

So, the coordinates of the point are $\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$.

What is the value of sin(π/4) and cos(π/4) on the unit circle?

To find the values of $\sin(\frac{\pi}{4})$ and $\cos(\frac{\pi}{4})$ on the unit circle, we use the coordinates of the point on the unit circle corresponding to the angle $\frac{\pi}{4}$.

The angle $\frac{\pi}{4}$ radians corresponds to 45 degrees. On the unit circle, the coordinates of this angle are $\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$.

Therefore, $$\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$$ and $$\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$$

How to Learn the Unit Circle

$$\text{To learn the unit circle, start by understanding that it is a circle with a radius of 1 centered at the origin (0,0).}$$

$$\text{1. Memorize the key angles: 0°, 30°, 45°, 60°, 90°, and their equivalents in radians.}$$

$$\text{2. Know the coordinates of the points where these angles intersect the unit circle. For example, (1,0) at 0°, (0,1) at 90°.}$$

$$\text{3. Understand the sine and cosine functions, which give the y and x coordinates of these points, respectively.}$$