Unit Circle

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

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

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

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

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

Find the exact values of the trigonometric functions for an angle of 7π/6 radians on the unit circle

We need to find the exact values of sine, cosine, and tangent for the angle $\frac{7\pi}{6}$ radians.

1. Find the reference angle:

The reference angle for $\frac{7\pi}{6}$ is $\pi – \frac{7\pi}{6} = \frac{\pi}{6}$.

2. Determine the signs in the third quadrant:

In the third quadrant, sine and cosine are negative, and tangent is positive.

3. Use the reference angle to find the values:

$\sin\left(\frac{7\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right) = -\frac{1}{2}$

$\cos\left(\frac{7\pi}{6}\right) = -\cos\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2}$

$\tan\left(\frac{7\pi}{6}\right) = \frac{\sin\left(\frac{7\pi}{6}\right)}{\cos\left(\frac{7\pi}{6}\right)} = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

Finding Reference Angle for Angles not on the Unit Circle

To find the reference angle for an angle not on the unit circle, we first need to understand the definition of a reference angle. A reference angle is the acute angle formed by the terminal side of the given angle and the horizontal axis. The reference angle is always between 0° and 90°, and it is always positive.

Let’s consider an example: Find the reference angle for the angle 250°.

Step 1: Determine the quadrant in which the given angle lies. Since 250° is between 180° and 270°, it lies in the third quadrant.

Step 2: Use the formula for the reference angle in the third quadrant:

$$ \theta_{reference} = \theta – 180° $$

For our example:

$$ \theta_{reference} = 250° – 180° = 70° $$

Therefore, the reference angle for 250° is 70°.

What are the sine, cosine, and tangent of the angle π/3 on the unit circle?

First, locate the angle $\frac{\pi}{3}$ on the unit circle. This angle corresponds to 60 degrees.

The coordinates of the point on the unit circle at this angle are $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.

Thus, the cosine of $\frac{\pi}{3}$ is the x-coordinate, which is $\frac{1}{2}$:

$$\cos \frac{\pi}{3} = \frac{1}{2}$$

The sine of $\frac{\pi}{3}$ is the y-coordinate, which is $\frac{\sqrt{3}}{2}$:

$$\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$$

The tangent is given by the ratio of the sine to the cosine:

$$\tan \frac{\pi}{3} = \frac{\sin \frac{\pi}{3}}{\cos \frac{\pi}{3}} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}$$

What are the coordinates of the point on the unit circle at an angle of π/3 radians?

Given an angle of $\frac{\pi}{3}$ radians, we want to find the coordinates of the corresponding point on the unit circle.

The unit circle has a radius of 1, and the coordinates of any point on the unit circle can be found using the cosine and sine of the angle.

Therefore, the x-coordinate is $\cos(\frac{\pi}{3})$ and the y-coordinate is $\sin(\frac{\pi}{3})$.

We know from trigonometric values:

$$\cos(\frac{\pi}{3}) = \frac{1}{2}$$

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

Thus, the coordinates are:

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