Homework

Find the cosine of an angle of \(\frac{\pi}{3}\) radians on the unit circle

To find the cosine of an angle of $\frac{\pi}{3}$ radians on the unit circle, we need to look at the coordinates of the point where the terminal side of the angle intersects the unit circle.

On the unit circle, the coordinates are given by $(\cos \theta, \sin \theta)$ where $\theta$ is the angle in radians.

For $\theta = \frac{\pi}{3}$, the coordinates are $(\frac{1}{2}, \frac{\sqrt{3}}{2})$.

Thus, the cosine of $\frac{\pi}{3}$ is $\frac{1}{2}$.

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

Find the coordinates of a point and the corresponding angle on the unit circle given the sine value, and prove if the cosine value meets the trigonometric identity

Given the sine value $\sin(\theta) = \frac{3}{5}$ on the unit circle, find the coordinates $(x,y)$ of the point and the angle $\theta$. Verify if the cosine value $\cos(\theta)$ satisfies the trigonometric identity.

Step 1: Use the Pythagorean identity:

$$\sin^2(\theta) + \cos^2(\theta) = 1$$

Step 2: Substitute $\sin(\theta) = \frac{3}{5}$ into the identity:

$$\left(\frac{3}{5}\right)^2 + \cos^2(\theta) = 1$$

$$\frac{9}{25} + \cos^2(\theta) = 1$$

Step 3: Solve for $\cos^2(\theta)$:

$$\cos^2(\theta) = 1 – \frac{9}{25}$$

$$\cos^2(\theta) = \frac{16}{25}$$

Step 4: Determine $\cos(\theta)$:

$$\cos(\theta) = \pm \frac{4}{5}$$

Step 5: Verify the coordinates:

The coordinates are $(\pm \frac{4}{5}, \frac{3}{5})$ for $\theta = \arcsin(\frac{3}{5})$.

Find the value of cos(θ) on the unit circle when θ = 60°

To solve for $\cos(60°)$, we can use the unit circle, where $\theta$ represents the angle from the positive x-axis.

On the unit circle, the coordinates of a point at an angle $\theta$ are $(\cos(\theta), \sin(\theta))$.

For $\theta = 60°$, the coordinates are $(\frac{1}{2}, \frac{\sqrt{3}}{2})$. Hence, $\cos(60°) = \frac{1}{2}$.

$$\cos(60°) = \frac{1}{2}$$

Find the cosine and sine values of an angle in the unit circle

Given an angle of \( \frac{5\pi}{4} \) radians, determine the cosine and sine values using the unit circle.

First, locate the angle \( \frac{5\pi}{4} \) on the unit circle. This angle is in the third quadrant where both sine and cosine values are negative.

The reference angle for \( \frac{5\pi}{4} \) is \( \frac{\pi}{4} \). In the unit circle, the sine and cosine of \( \frac{\pi}{4} \) are both \( \frac{\sqrt{2}}{2} \).

Since \( \frac{5\pi}{4} \) is in the third quadrant, the values become negative:

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

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

Find the cosecant of the angle π/6 on the unit circle

To find the cosecant of the angle $\frac{\pi}{6}$ on the unit circle, we first need to find the sine of $\frac{\pi}{6}$.

On the unit circle, the sine of $\frac{\pi}{6}$ is $\frac{1}{2}$.

The cosecant is the reciprocal of the sine.

So, the cosecant of $\frac{\pi}{6}$ is:

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

Conversion Problem on the Unit Circle

$$\text{Given that } \theta = \frac{5\pi}{4} \text{ radians}, \text{ convert this angle to degrees and then determine the coordinates of the corresponding point on the unit circle.}$$

$$\text{To convert radians to degrees, use the formula:}$$

$$\theta_{deg} = \theta_{rad} \times \frac{180^{\circ}}{\pi}$$

$$\theta_{deg} = \frac{5\pi}{4} \times \frac{180^{\circ}}{\pi}$$

$$\theta_{deg} = 225^{\circ}$$

$$\text{Next, find the coordinates on the unit circle for } 225^{\circ}. \text{ This corresponds to the angle } 225^{\circ} \text{ or } \frac{5\pi}{4} \text{ radians.}$$

$$\cos(225^{\circ}) = \cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$$

$$\sin(225^{\circ}) = \sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$$

$$\text{Therefore, the coordinates are: } \left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$$

Find the real part of the complex number $z$ on the unit circle given by $z = e^{i\theta}$ and $\theta = \frac{\pi}{4}$

We are given the complex number $z$ on the unit circle:

$$z = e^{i\theta}$$

For $\theta = \frac{\pi}{4}$, we have:

$$z = e^{i\frac{\pi}{4}}$$

By Euler’s formula, $e^{i\theta} = \cos \theta + i \sin \theta$, so:

$$e^{i\frac{\pi}{4}} = \cos \frac{\pi}{4} + i \sin \frac{\pi}{4}$$

We know $\cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$, thus:

$$e^{i\frac{\pi}{4}} = \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}$$

Therefore, the real part of $z$ is:

$$\boxed{\frac{\sqrt{2}}{2}}$$

Identifying Quadrants on the Unit Circle

$$Identify\ the\ quadrant\ in\ which\ the\ angle\ \theta = 5\pi/4\ \text{radians}\ lies\ on\ the\ unit\ circle.$$

$$To\ determine\ the\ quadrant\ of\ \theta = 5\pi/4:\ $$

$$1. \text{Convert\ the\ angle\ to\ degrees\ for\ better\ understanding:}$$

$$\theta = \frac{5\cdot180}{4} = 225^{\circ}\ $$

$$2. \text{Analyze\ the\ degree\ measure:}$$

$$0^{\circ} \leq 225^{\circ} \leq 360^{\circ}\ $$

$$225^{\circ} \text{lies\ between\ 180^{\circ}\ (negative\ x-axis)\ and\ 270^{\circ}\ (negative\ y-axis),\ which\ is\ the\ Third\ Quadrant.}$$

$$Therefore,\ \theta = 5\pi/4\ \text{radians\ lies\ in\ the\ Third\ Quadrant.}$$

Find the sine, cosine, and tangent of an angle using the unit circle

To find the sine, cosine, and tangent of an angle $\theta$ on the unit circle, use the following steps:

1. Identify the coordinates $(x, y)$ on the unit circle corresponding to $\theta$.

2. The $x$-coordinate is $\cos(\theta)$.

3. The $y$-coordinate is $\sin(\theta)$.

4. The tangent of the angle is $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$.

For example, consider $\theta = \frac{\pi}{4}$:

1. The coordinates are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.

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

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

4. $\tan(\frac{\pi}{4}) = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$.

Find the coordinates of the point on the unit circle where the angle in radians is 7π/6

To find the coordinates of the point on the unit circle at the angle $\frac{7\pi}{6}$, we use the cosine and sine functions.

1. The angle $\frac{7\pi}{6}$ is in the third quadrant where both cosine and sine are negative.

2. The reference angle for $\frac{7\pi}{6}$ is $\frac{\pi}{6}$.

3. The cosine and sine of $\frac{\pi}{6}$ are $\frac{\sqrt{3}}{2}$ and $\frac{1}{2}$, respectively.

4. Therefore, the coordinates are:

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