Math

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

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

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

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

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

Find the sine and cosine values for \( \frac{\pi}{4} \) using the unit circle

To find the sine and cosine values for $\frac{\pi}{4}$ using the unit circle, we use the coordinates of the point corresponding to that angle on the circle.

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

Therefore:

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

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

Calculate the exact value of tan(-π/6) using the unit circle and verify by applying trigonometric identities

Using the unit circle, first note that $-\frac{\pi}{6}$ is equivalent to $-30^\circ$. On the unit circle, this angle corresponds to the coordinates $\left( \frac{\sqrt{3}}{2}, -\frac{1}{2} \right)$.

Therefore, the value of $\tan(-\frac{\pi}{6})$ is given by the ratio of the y-coordinate to the x-coordinate:

$$ \tan(-\frac{\pi}{6}) = \frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} $$

Verification using trigonometric identities can be done by noting that $\tan(-x) = -\tan(x)$. Hence,

$$ \tan(-\frac{\pi}{6}) = -\tan(\frac{\pi}{6}) = -\frac{\sqrt{3}}{3} $$

Find the exact values of sin(225°) and cos(225°) using the unit circle

To find the exact values of $\sin(225°)$ and $\cos(225°)$, we first locate $225°$ on the unit circle.

$225°$ is in the third quadrant. The reference angle is $225° – 180° = 45°$. In the third quadrant, the sine and cosine of the reference angle are both negative.

Therefore, $\sin(225°) = -\sin(45°) = – \frac{\sqrt{2}}{2}$ and $\cos(225°) = -\cos(45°) = -\frac{\sqrt{2}}{2}$.

Thus, the exact values are:

$$\sin(225°) = -\frac{\sqrt{2}}{2}$$

$$\cos(225°) = -\frac{\sqrt{2}}{2}$$

Find the coordinates of the point on the unit circle where the angle is π/10 radians

To find the coordinates of the point on the unit circle at an angle of $\frac{\pi}{10}$ radians, we use the cosine and sine functions:

$$x = \cos\left(\frac{\pi}{10}\right)$$

$$y = \sin\left(\frac{\pi}{10}\right)$$

Therefore, the coordinates are:

$$\left( \cos\left(\frac{\pi}{10}\right), \sin\left(\frac{\pi}{10}\right) \right)$$