Unit Circle

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

$$\text{The angle } \frac{\pi}{3} \text{ is equivalent to } 60^{\circ}.$$

$$\text{On the unit circle, the coordinates for } 60^{\circ} \text{ are } \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right).$$

$$\sin(\frac{\pi}{3}) \text{ is the y-coordinate, which is } \frac{\sqrt{3}}{2}.$$

Find the sine and cosine of 𝜋/6 radians on the unit circle

To find the sine and cosine of $\frac{\pi}{6}$ radians on the unit circle, we need to recall the standard angle values:

At $\frac{\pi}{6}$ radians:

$$cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}$$

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

Thus, the cosine of $\frac{\pi}{6}$ radians is $\frac{\sqrt{3}}{2}$, and the sine of $\frac{\pi}{6}$ radians is $\frac{1}{2}$.

Find the coordinates and trigonometric values for an angle on the unit circle

Consider an angle $ \theta = \frac{7\pi}{6} $ on the unit circle. We need to find the coordinates of the point on the unit circle corresponding to this angle, as well as the sine and cosine values.

First, identify the reference angle: $$ \theta_{ref} = \pi – \frac{7\pi}{6} = \frac{\pi}{6} $$

Next, find the coordinates for the reference angle $ \frac{\pi}{6} $:

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

Since $ \theta = \frac{7\pi}{6} $ is in the third quadrant, both sine and cosine are negative:

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

How to Remember the Unit Circle Fast

$$\text{To remember the unit circle, focus on key angles and their coordinates. Start with } 0^\circ, 30^\circ, 45^\circ, 60^\circ, \text{ and } 90^\circ.$$

$$\text{For example, at } 0^\circ, \text{ the coordinates are } (1, 0).$$

$$\text{At } 30^\circ, \text{ the coordinates are } \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right).$$

$$\text{At } 45^\circ, \text{ the coordinates are } \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right).$$

$$\text{At } 60^\circ, \text{ the coordinates are } \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right).$$

$$\text{At } 90^\circ, \text{ the coordinates are } (0, 1).$$

$$\text{Memorize these points, and use symmetry to fill in the rest of the circle.}$$

Find the value of tan(θ) on the unit circle for θ = 5π/4

To find the value of $ \tan(\theta) $ on the unit circle for $ \theta = \frac{5\pi}{4} $, we first determine the coordinates of the point on the unit circle that corresponds to this angle.

The angle $ \frac{5\pi}{4} $ is in the third quadrant, where both sine and cosine values are negative. The reference angle is $ \pi/4 $.

The coordinates for $ \theta = \frac{5\pi}{4} $ are $ (-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}) $.

Since $ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $, we have:

$$ \tan\left(\frac{5\pi}{4}\right) = \frac{\sin\left(\frac{5\pi}{4}\right)}{\cos\left(\frac{5\pi}{4}\right)} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1 $$

What does sin represent on the unit circle?

On the unit circle, the sine of an angle represents the y-coordinate of the point on the unit circle that corresponds to that angle.

$$\text{Given an angle } \theta, \text{the coordinates of the point P on the unit circle are } (\cos(\theta), \sin(\theta)).$$

This means:

$$\sin(\theta) = y.$$

For example, if \(\theta = 30^\circ\):

$$\sin(30^\circ) = \frac{1}{2}.$$

Find the value of sec(θ) if point P(1/2, √3/2) lies on the unit circle

To find $\sec(\theta)$, we need to know $\cos(\theta)$. Given the coordinates on the unit circle, $\cos(\theta) = x$-coordinate of point $P$.

Here, $x = \frac{1}{2}$. Therefore, $\cos(\theta) = \frac{1}{2}$.

Recall that $\sec(\theta) = \frac{1}{\cos(\theta)}$.

Thus, $\sec(\theta) = \frac{1}{\frac{1}{2}} = 2$.

Therefore, $\sec(\theta) = 2$.

Find the coordinates of the point where the terminal side of a 225-degree angle intersects the unit circle

Given an angle of $225^{\circ}$, we first convert it to radians:

$$225^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{5\pi}{4} \text{ radians}$$

The angle $\frac{5\pi}{4}$ is in the third quadrant, where both sine and cosine are negative. The reference angle is $225^{\circ} – 180^{\circ} = 45^{\circ}$. Since $\sin(45^{\circ}) = \frac{\sqrt{2}}{2}$ and $\cos(45^{\circ}) = \frac{\sqrt{2}}{2}$, we have:

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

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

Thus, the coordinates are:

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

What Does Sin Represent on the Unit Circle?

On the unit circle, the function $\sin$ represents the y-coordinate of a point on the circle. The unit circle is defined as a circle with a radius of 1, centered at the origin of the coordinate system.

First, consider a point on the unit circle defined by an angle $\theta$, measured in radians from the positive x-axis. This point can be represented as $(\cos(\theta), \sin(\theta))$.

Because the radius of the unit circle is 1, the coordinates $(\cos(\theta), \sin(\theta))$ correspond to the horizontal and vertical distances from the origin.

Therefore,

$$\sin(\theta)$$

represents the vertical distance from the x-axis to the point on the unit circle. For example, if $\theta = \frac{\pi}{6}$ (30 degrees), the point on the unit circle is $(\frac{\sqrt{3}}{2}, \frac{1}{2})$. Thus, $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$.

Find the coordinates of the points where the line y = x intersects the unit circle

We start with the unit circle equation:

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

Substituting $y = x$, we get:

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

$$2x^2 = 1$$

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

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

Since $y = x$, the coordinates are:

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

and

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