Unit Circle

What is the value of sin(π/4) using the unit circle?

To find the value of $ \sin(\frac{\pi}{4}) $ using the unit circle, we need to identify the coordinates of the point on the unit circle corresponding to $ \frac{\pi}{4} $ radians.

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

Since $ \sin(\theta) $ is equal to the y-coordinate of the point on the unit circle, we have

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

Find the sine of an angle in radians on the unit circle

To find the sine of an angle on the unit circle, we need to find the y-coordinate of the point where the terminal side of the angle intersects the unit circle.

Let’s consider the angle \( \frac{5\pi}{6} \).

First, we need to identify the reference angle, which is the acute angle formed with the x-axis. For \( \frac{5\pi}{6} \), the reference angle is \( \pi – \frac{5\pi}{6} = \frac{\pi}{6} \).

On the unit circle, the coordinates of the point corresponding to \( \frac{\pi}{6} \) are \( ( \frac{\sqrt{3}}{2}, \frac{1}{2} ) \).

Since \( \frac{5\pi}{6} \) is in the second quadrant, the sine value (y-coordinate) remains positive.

Therefore, the sine of \( \frac{5\pi}{6} \) is:

$$ \sin \left( \frac{5\pi}{6} \right) = \frac{1}{2} $$

Find the Coordinates of a Point on the Unit Circle

Given a unit circle, find the coordinates of the point where the terminal side of an angle $\theta = \frac{5\pi}{4}$ intersects the circle.

Solution:

To find the coordinates of the point where the terminal side of an angle $\theta = \frac{5\pi}{4}$ intersects the unit circle, we use the following formulas:
$x = \cos(\theta)$
$y = \sin(\theta)$

For $\theta = \frac{5\pi}{4}$:
$x = \cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$
$y = \sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$

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

Find the coordinates of a point on the negative unit circle

To find the coordinates of a point on the negative unit circle, we need to remember that the equation for a unit circle is $x^2 + y^2 = 1$. For a point on the negative unit circle, both x and y values will be negative.

Let’s take an example where $x = -\frac{1}{2}$. So,

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

Substituting $x = -\frac{1}{2}$ into the equation, we get:

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

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

$$ y^2 = 1 – \frac{1}{4} $$

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

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

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

Therefore, the coordinates of the point are: $(-\frac{1}{2}, -\frac{\sqrt{3}}{2})$.

Determine the coordinates of the point on the unit circle corresponding to an angle of 5π/4 radians

To find the coordinates of the point on the unit circle corresponding to an angle of $$\frac{5\pi}{4}$$ radians, we can use the unit circle properties.

The angle $$\frac{5\pi}{4}$$ is located in the third quadrant of the unit circle. The reference angle for $$\frac{5\pi}{4}$$ is $$\pi – \frac{5\pi}{4} = \frac{\pi}{4}$$, which corresponds to a 45-degree angle.

For a point in the third quadrant, both the sine (y-coordinate) and cosine (x-coordinate) values will be negative. The coordinates of a 45-degree angle on the unit circle are (sqrt(2)/2, sqrt(2)/2). Therefore, the coordinates for the angle $$\frac{5\pi}{4}$$ will be:

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

Determine the sine and cosine of an angle in a unit circle

Given an angle of $\frac{\pi}{4}$ radians, determine the coordinates on the unit circle.

In a unit circle, the coordinates for $\frac{\pi}{4}$ are $(\cos(\frac{\pi}{4}), \sin(\frac{\pi}{4}))$.

Using the known values:

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

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

Therefore, the coordinates are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.

What is the y-coordinate of the point on the unit circle at an angle of π/3?

To find the y-coordinate of the point on the unit circle at an angle of $\frac{\pi}{3}$, we use the sine function.

The sine of $\frac{\pi}{3}$ is $\frac{\sqrt{3}}{2}$.

Therefore, the y-coordinate is $$\frac{\sqrt{3}}{2}$$.

Determine the sine and cosine of the angle π/4 on the unit circle

To find the sine and cosine of the angle $\frac{\pi}{4}$ on the unit circle, we use the definitions of sine and cosine for the unit circle.

For an angle $\theta$ in the unit circle, $\cos(\theta)$ is the x-coordinate and $\sin(\theta)$ is the y-coordinate of the corresponding point.

At $\theta = \frac{\pi}{4}$, the coordinates are known to be $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$.

Thus,

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

and

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

Given a point on the unit circle at an angle of 3π/4 radians, find the coordinates of this point and verify the trigonometric identities for sine and cosine at this angle

To solve this problem, we first need to understand the unit circle and the angle $\frac{3\pi}{4}$ radians.

On the unit circle, the angle $\frac{3\pi}{4}$ is located in the second quadrant where sine is positive and cosine is negative. The reference angle for $\frac{3\pi}{4}$ radians is $\frac{\pi}{4}$ radians.

For the reference angle $\frac{\pi}{4}$, the sine and cosine values are both equal to $\frac{\sqrt{2}}{2}$.

Therefore, at $\frac{3\pi}{4}$ radians, the sine is positive, and the cosine is negative:

$$\sin\left( \frac{3\pi}{4} \right) = \frac{\sqrt{2}}{2}$$

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

So, the coordinates of the point at $\frac{3\pi}{4}$ radians on the unit circle are:

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

To verify the trigonometric identities, we can check:

$$\sin^2\left( \frac{3\pi}{4} \right) + \cos^2\left( \frac{3\pi}{4} \right) = \left( \frac{\sqrt{2}}{2} \right)^2 + \left( -\frac{\sqrt{2}}{2} \right)^2 = \frac{1}{2} + \frac{1}{2} = 1$$

Thus, the identities are verified.

Find the value of \( \theta \) if \( \tan(\theta) = 2 \) and \( \theta \) is in the second quadrant Then, calculate the coordinates of the corresponding point on the unit circle

We know that \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). In the second quadrant, sine is positive and cosine is negative. Let us find \( \theta \) such that \( \tan(\theta) = 2 \). The reference angle \( \theta_r \) is given by:

$$ \theta_r = \arctan(2) $$

Since \( \theta \) is in the second quadrant, the angle \( \theta \) is:

$$ \theta = \pi – \theta_r = \pi – \arctan(2) $$

Next, to find the coordinates of the corresponding point on the unit circle, we use the unit circle property \((\cos(\theta), \sin(\theta))\). First we find \( \sin(\theta) \) and \( \cos(\theta) \) using:

$$ \sin(\theta) = \frac{2}{\sqrt{1 + 2^2}} = \frac{2}{\sqrt{5}} \quad \text{and} \quad \cos(\theta) = -\frac{1}{\sqrt{1 + 2^2}} = -\frac{1}{\sqrt{5}} $$

Therefore, the coordinates are:

$$ \left( -\frac{1}{\sqrt{5}}, \frac{2}{\sqrt{5}} \right) $$