Unit Circle

Determine the values of tan(θ) on the unit circle where tan(θ) = 1 or tan(θ) = -1

First, note that $ \tan(\theta) = 1 $ when $ \theta = \frac{\pi}{4} $ or $ \theta = \frac{5\pi}{4} $ on the unit circle. Also, $ \tan(\theta) = -1 $ when $ \theta = \frac{3\pi}{4} $ or $ \theta = \frac{7\pi}{4} $. Therefore, the angles where $ \tan(\theta) = 1 $ or $ \tan(\theta) = -1 $ are:

$$ \theta = \frac{\pi}{4}, \frac{5\pi}{4}, \frac{3\pi}{4}, \frac{7\pi}{4} $$

Find the value of arctan(sin(3π/4))

To find the value of $ \arctan(\sin(\frac{3\pi}{4})) $, we first need to find the value of $ \sin(\frac{3\pi}{4}) $.

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

Now, we need to determine the value of $ \arctan(\frac{\sqrt{2}}{2}) $.

Since $ \arctan(x) $ is the inverse of $ \tan(x) $, we seek an angle $ \theta $ such that:

$$ \tan(\theta) = \frac{\sqrt{2}}{2} $$

One such angle is $ \theta = \frac{\pi}{4} $, but considering the range of $ \arctan $, the solution is:

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

Find the coordinates of the point on the unit circle where the angle with the positive x-axis is pi/3

The unit circle is defined as a circle with radius 1 centered at the origin. The coordinates of any point on the unit circle can be given by $(\cos(\theta), \sin(\theta))$ where $\theta$ is the angle with the positive $x$-axis.

Given that $\theta = \frac{\pi}{3}$:

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

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

Thus, the coordinates are:

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

Find the coordinates of the point where the terminal side of an angle $\theta$ intersects the unit circle, given that $\theta = \frac{5\pi}{6}$

To determine the coordinates where the terminal side of $\theta = \frac{5\pi}{6}$ intersects the unit circle:

First, recall that on the unit circle, the coordinates are given by $(\cos(\theta), \sin(\theta))$.

Calculate the cosine and sine values:

$$ \cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2} $$

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

Thus, the coordinates are:

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

Determine the exact values of sine and cosine for an angle of 5π/6 in the unit circle

To determine the exact values of $\sin(\frac{5\pi}{6})$ and $\cos(\frac{5\pi}{6})$, we use the unit circle.

For the angle $\frac{5\pi}{6}$, it is in the second quadrant where sine is positive and cosine is negative. The reference angle for $\frac{5\pi}{6}$ is:

$$ \pi – \frac{5\pi}{6} = \frac{\pi}{6} $$

The values for sine and cosine at $\frac{\pi}{6}$ are known:

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

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

Since $\frac{5\pi}{6}$ is in the second quadrant:

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

$$ \cos(\frac{5\pi}{6}) = – \cos(\frac{\pi}{6}) = – \frac{\sqrt{3}}{2} $$

Given a point on the unit circle, find the corresponding angle and verify the coordinates

Let

Determine the value of tan(θ) at θ = 3π/4 using the unit circle chart

To determine the value of $ \tan(\theta) $ at $ \theta = \frac{3\pi}{4} $, we use the unit circle chart. The angle $ \frac{3\pi}{4} $ is in the second quadrant, where the reference angle is $ \frac{\pi}{4} $. In this quadrant, the tangent value is negative.

Since $ \tan(\frac{\pi}{4}) = 1 $,

$$ \tan(\frac{3\pi}{4}) = -1 $$

Therefore, the value of $ \tan(\frac{3\pi}{4}) $ is:

$$ \boxed{-1} $$

Find the coordinates of a point on the unit circle at angle pi/3

The unit circle has a radius of 1. The coordinates of a point on the circle at angle $ \frac{\pi}{3} $ are given by:

$$ (\cos(\frac{\pi}{3}), \sin(\frac{\pi}{3})) $$

Therefore,

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

and

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

Thus, the coordinates are:

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

Determine the values of sin(θ), cos(θ), and tan(θ) for θ in the second quadrant of the unit circle

In the second quadrant, the angle $ \theta $ ranges from $ \frac{\pi}{2} $ to $ \pi $. Here, $ \sin(\theta) $ is positive, $ \cos(\theta) $ is negative, and $ \tan(\theta) $ is negative.

Using the unit circle, for $ \theta = \frac{2\pi}{3} $:

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

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

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

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

To find the value of $ \sin\left(\frac{\pi}{3}\right) $ on the unit circle, we look at the reference angle for $ \frac{\pi}{3} $.

The reference angle is $ 60^{\circ} $, and the sine value for $ 60^{\circ} $ is:

$$ \sin\left(60^{\circ}\right) = \frac{\sqrt{3}}{2} $$

Therefore:

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