Homework

Find the value of tan(θ) given a point on the unit circle

Given a point on the unit circle at coordinates $(x, y)$, find the value of $\tan(\theta)$ where $\theta$ is the angle formed by the radius connecting the point to the origin.

Using the definition of tangent in the unit circle:

$$\tan(\theta) = \frac{y}{x}$$

For example, if the point on the unit circle is $(\frac{1}{2}, \frac{\sqrt{3}}{2})$, then:

$$\tan(\theta) = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}$$

Determine the Quadrant on a Unit Circle

To determine the quadrant of the angle \( \theta \) on the unit circle, we need to understand the angle’s position in relation to the x-axis and y-axis.

Consider the angle \( \theta = 150^{\circ} \).

Step 1: Convert the angle to radians if needed. \( 150^{\circ} = \frac{5\pi}{6} \) radians.

Step 2: Identify the reference angle and its position. Since \( 150^{\circ} \) is between \( 90^{\circ} \) and \( 180^{\circ} \), it lies in the second quadrant.

Answer: The quadrant of \( 150^{\circ} \) is Quadrant II.

Find the value of cosine for an angle on the unit circle

Let’s find the value of $\cos(\frac{\pi}{4})$ on the unit circle.

The angle $\frac{\pi}{4}$ is equivalent to 45 degrees.

On the unit circle, the coordinates of the point at an angle of $\frac{\pi}{4}$ are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.

Thus, the cosine of $\frac{\pi}{4}$ is $$\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$$.

What is the cosine and sine of the angle π/4 on the unit circle?

To find the cosine and sine of the angle $\frac{\pi}{4}$ on the unit circle, we need to recall the coordinates of the point where the terminal side of the angle intersects the unit circle.

For the angle $\frac{\pi}{4}$, both the x-coordinate (cosine) and y-coordinate (sine) are equal. Since the unit circle has a radius of 1, we use the fact that $\cos(\theta) = \sin(\theta) = \frac{\sqrt{2}}{2}$ for this specific angle. Therefore,

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

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

Find the cotangent of \( \frac{\pi}{4} \) on the unit circle

To find the cotangent of $ \frac{\pi}{4} $ on the unit circle, we use the definition of cotangent in terms of sine and cosine.

$$ \cot \theta = \frac{\cos \theta}{\sin \theta} $$

For $ \theta = \frac{\pi}{4} $, both $ \sin \frac{\pi}{4} $ and $ \cos \frac{\pi}{4} $ are $ \frac{\sqrt{2}}{2} $.

Therefore,

$$ \cot \frac{\pi}{4} = \frac{\cos \frac{\pi}{4}}{\sin \frac{\pi}{4}} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 $$

The cotangent of $ \frac{\pi}{4} $ is 1.

Find the value of sec(θ) using the unit circle when θ = 2π/3, and verify the result using three different methods

First, we find the coordinates of the point on the unit circle corresponding to $\theta = \frac{2\pi}{3}$.

The coordinates are $\left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right)$.

Since $\sec(\theta) = \frac{1}{\cos(\theta)}$, we have:

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

Verification using the Pythagorean identity:

$$\sec^2(\theta) = 1 + \tan^2(\theta)$$

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

$$\sec^2\left(\frac{2\pi}{3}\right) = 1 + 3 = 4$$

$$\sec\left(\frac{2\pi}{3}\right) = \pm 2 = -2.$$

Find the sine and cosine values for the angle 5π/6 using the unit circle

First, locate the angle $\frac{5\pi}{6}$ on the unit circle.

The angle $\frac{5\pi}{6}$ is in the second quadrant.

In the second quadrant, sine is positive and cosine is negative.

The reference angle for $\frac{5\pi}{6}$ is $\pi – \frac{5\pi}{6} = \frac{\pi}{6}$.

From the unit circle, $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$ and $\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$.

Thus, $\sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}$ and $\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}$.

Find the value of tan for given angles on the unit circle

Consider the angle $\theta = \frac{3\pi}{4}$ on the unit circle.

First, determine the reference angle. The reference angle for $\frac{3\pi}{4}$ is $\frac{\pi}{4}$.

Since $\frac{3\pi}{4}$ is in the second quadrant, tangent is negative.

We know $\tan \frac{\pi}{4} = 1$, so:

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

Cosine Values on the Unit Circle

Consider the point $P(\frac{1}{2}, \frac{\sqrt{3}}{2})$ on the unit circle. Determine the cosine of the angle $\theta$ corresponding to this point.

Solution:

On the unit circle, the coordinates of a point $P(x, y)$ correspond to $(\cos \theta, \sin \theta)$. Given the coordinates $P(\frac{1}{2}, \frac{\sqrt{3}}{2})$, we can identify that $\cos \theta = \frac{1}{2}$.

Thus, the cosine of the angle $\theta$ is:

$$\cos \theta = \frac{1}{2}$$