Math

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:

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

Find the cotangent value and corresponding angle on the unit circle

We need to find the angle $\theta$ in the unit circle such that $\cot(\theta) = \sqrt{3}$.

Step 1: Recall that $\cot(\theta) = \frac{1}{\tan(\theta)}$ and $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$.

Step 2: Set up the equation $\frac{1}{\tan(\theta)} = \sqrt{3}$, which then gives $\tan(\theta) = \frac{1}{\sqrt{3}}$.

Step 3: Recall that $\tan(30^\circ) = \tan(\frac{\pi}{6}) = \frac{1}{\sqrt{3}}$.

Therefore, the angle $\theta = 30^\circ$ or $\theta = \frac{\pi}{6}$ in radians, since $\cot(30^\circ) = \sqrt{3}$.