Unit Circle

Given a point on the unit circle at an angle of 5π/4 radians, find the coordinates of this point Then, if the unit circle is flipped about the y-axis, determine the new coordinates of the original point after the flip

First, find the coordinates of the point on the unit circle at $\frac{5\pi}{4}$ radians. This point can be represented as:

$$ (\cos(\frac{5\pi}{4}), \sin(\frac{5\pi}{4})) $$

We know that:

$$ \cos(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2} $$

$$ \sin(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2} $$

Thus, the coordinates at $\frac{5\pi}{4}$ radians are:

$$ (-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}) $$

Next, when the unit circle is flipped about the y-axis, the x-coordinate of the point changes sign, but the y-coordinate remains the same. Therefore, the new coordinates after the flip are:

$$ (\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}) $$

Find the values of cotangent for a given angle on the unit circle and verify their consistency

To find the values of $\cot(\theta)$ for $\theta = \frac{3\pi}{4}$ on the unit circle, we start by identifying the coordinates of this angle on the unit circle.

The coordinates for $\theta = \frac{3\pi}{4}$ are $\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$. The cotangent function is defined as the cosine divided by the sine of the angle: $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$.

Therefore,

$$\cot\left(\frac{3\pi}{4}\right) = \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = -1$$

The value of $\cot\left(\frac{3\pi}{4}\right)$ is -1.

What is the cosine of the angle at 2π/3 on the unit circle?

First, recognize that the angle $ \frac{2\pi}{3} $ is in the second quadrant.

In the unit circle, the cosine function is negative in the second quadrant.

The reference angle for $ \frac{2\pi}{3} $ is $ \pi – \frac{2\pi}{3} = \frac{\pi}{3} $.

We know that $ \cos\left( \frac{\pi}{3} \right) = \frac{1}{2} $.

Therefore, the cosine of $ \frac{2\pi}{3} $ is:

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

Determine the Tangent Slope at a Given Point on the Unit Circle

Let the given point on the unit circle be $(a, b)$, where $a^2 + b^2 = 1$. We need to determine the slope of the tangent line at this point.

The equation of the unit circle is given by:

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

To find the slope of the tangent line at $(a, b)$, we first implicitly differentiate both sides of the equation with respect to $x$:

$$2x + 2y\frac{dy}{dx} = 0$$

Solving for $\frac{dy}{dx}$:

$$\frac{dy}{dx} = -\frac{x}{y}$$

Substituting the point $(a, b)$ into the derivative:

$$\frac{dy}{dx}\bigg|_{(a,b)} = -\frac{a}{b}$$

Therefore, the slope of the tangent line at the point $(a, b)$ is $-\frac{a}{b}$.

Find the value of tan(4π/3) using the unit circle

$$To \ find \ the \ value \ of \ tan(\frac{4\pi}{3}) \ using \ the \ unit \ circle, \ we \ first \ need \ to \ determine \ the \ reference \ angle. \ $$

$$The \ angle \ \frac{4\pi}{3} \ is \ in \ the \ third \ quadrant, \ and \ its \ reference \ angle \ is \ \frac{4\pi}{3} – \pi = \frac{\pi}{3}. \ $$

$$Using \ the \ unit \ circle, \ the \ coordinates \ for \ \frac{\pi}{3} \ are \ (\frac{1}{2}, \ \frac{\sqrt{3}}{2}). \ $$

$$Since \ the \ angle \ \frac{4\pi}{3} \ is \ in \ the \ third \ quadrant, \ both \ x \ and \ y \ coordinates \ are \ negative: \ (-\frac{1}{2}, \ -\frac{\sqrt{3}}{2}). \ $$

$$Finally, \ the \ tangent \ is \ the \ ratio \ of \ y \ to \ x: \ tan(\frac{4\pi}{3}) = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \sqrt{3}. \ $$

$$Therefore, \ tan(\frac{4\pi}{3}) = \sqrt{3}. \ $$

Finding the Location of -π/2 on a Unit Circle

To find the location of $-\pi/2$ on the unit circle, we start by understanding that angles are measured from the positive x-axis, and negative angles are measured clockwise.

For $-\pi/2$ radians, start from the positive x-axis and measure clockwise by $\pi/2$ radians (or 90 degrees). This brings us to the negative y-axis.

The coordinates of this point on the unit circle are $$(0, -1)$$.

So, $-\pi/2$ radians corresponds to the point (0, -1) on the unit circle.

Find the coordinates of a point on the unit circle that corresponds to a complex exponential representation

To find the coordinates of a point on the unit circle corresponding to $e^{i\theta}$ where $\theta = \frac{5\pi}{4}$, we use Euler’s formula:

$$e^{i\theta} = \cos(\theta) + i\sin(\theta)$$

Substituting $\theta = \frac{5\pi}{4}$:

$$e^{i\frac{5\pi}{4}} = \cos\left(\frac{5\pi}{4}\right) + i\sin\left(\frac{5\pi}{4}\right)$$

From the unit circle, we know:

$$\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$$

$$\sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$$

Thus, the coordinates are:

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

Calculate the coordinates and angles on the unit circle

To find the coordinates of the point on the unit circle corresponding to an angle of $\frac{5\pi}{4}$ radians, we use the trigonometric functions sine and cosine. For any angle $\theta$, the coordinates are given by:

$$ (\cos \theta, \sin \theta) $$

For $\theta = \frac{5\pi}{4}$, we have:

$$ \cos \left( \frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2} $$

$$ \sin \left( \frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2} $$

Thus, the coordinates of the point are:

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

What are the sine and cosine values for the angles 30°, 45°, and 60° on the unit circle?

To solve for the sine and cosine values for the angles 30°, 45°, and 60° on the unit circle, we need to refer to the specific values they correspond to:

For 30° (or π/6 radians):

$$\sin(30°) = \frac{1}{2}$$

$$\cos(30°) = \frac{\sqrt{3}}{2}$$

For 45° (or π/4 radians):

$$\sin(45°) = \frac{\sqrt{2}}{2}$$

$$\cos(45°) = \frac{\sqrt{2}}{2}$$

For 60° (or π/3 radians):

$$\sin(60°) = \frac{\sqrt{3}}{2}$$

$$\cos(60°) = \frac{1}{2}$$

Memorizing Key Angles and Coordinates on the Unit Circle

$$To memorize key angles and coordinates on the unit circle, start with the basic angles in degrees and radians. Recall that the unit circle has a radius of 1. $$

$$1. Identify the angles: 0°, 30°, 45°, 60°, 90°, and their corresponding radian measures: 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}. $$

$$2. Learn the coordinates: The coordinates for these angles are (1,0), (\frac{\sqrt{3}}{2}, \frac{1}{2}), (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}), (\frac{1}{2}, \frac{\sqrt{3}}{2}), and (0,1). $$

$$3. Use symmetry: The unit circle is symmetrical, so you can use the first quadrant to find coordinates in other quadrants by considering the signs of the x and y coordinates. $$