Math

Find the sine of the angle θ on the unit circle if θ = 30 degrees

To find the sine of $\theta$ on the unit circle, we can use the fact that $\sin(\theta)$ represents the y-coordinate of the point on the unit circle corresponding to the angle $\theta$.

For $\theta = 30^\circ$, we have:

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

Therefore, the sine of $30^\circ$ is $\frac{1}{2}$.

Find the coordinates on the unit circle for an angle of 135 degrees

To find the coordinates on the unit circle for an angle of $135^{\circ}$, we first convert degrees to radians.

$$135^{\circ} = \frac{135 \pi}{180} = \frac{3\pi}{4}$$

Next, we use the unit circle definitions for sine and cosine at $\frac{3 \pi}{4}$.

$$\cos \left( \frac{3\pi}{4} \right) = -\frac{\sqrt{2}}{2}$$

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

Thus, the coordinates are:

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

Determine the value of the trigonometric function for a specific angle

To find the value of the trigonometric function for a specific angle, we first need to identify the standard angle and then use the unit circle properties. Consider the angle $ \theta = \frac{5\pi}{4} $.

The reference angle is $ \frac{\pi}{4} $, and it lies in the third quadrant.

In the third quadrant, both sine and cosine values are negative. Therefore,

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

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

Thus,

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

Find the cosine values on the unit circle for specific angles

Let’s find the cosine values for angles 120°, 210°, and 330° on the unit circle.

First, convert the angles into radians:

$$120° = \frac{2\pi}{3}$$

$$210° = \frac{7\pi}{6}$$

$$330° = \frac{11\pi}{6}$$

Next, we use the unit circle to find the cosine values for each angle:

For $$\frac{2\pi}{3}$$, the cosine value is:

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

For $$\frac{7\pi}{6}$$, the cosine value is:

$$\cos \frac{7\pi}{6} = -\frac{\sqrt{3}}{2}$$

For $$\frac{11\pi}{6}$$, the cosine value is:

$$\cos \frac{11\pi}{6} = \frac{\sqrt{3}}{2}$$

Find the sine, cosine, and tangent of 150 degrees using the unit circle

First, convert 150 degrees to radians:

$$150^{\circ} = \frac{5\pi}{6} \text{ radians}$$

Next, identify the coordinates of the corresponding point on the unit circle:

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

Therefore,

$$\sin(150^{\circ}) = \frac{1}{2}$$

$$\cos(150^{\circ}) = -\frac{\sqrt{3}}{2}$$

To find the tangent:

$$\tan(150^{\circ}) = \frac{\sin(150^{\circ})}{\cos(150^{\circ})} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$$

Find the coordinates of a point on the unit circle where the angle is 45 degrees

To find the coordinates of a point on the unit circle at an angle of $45^\circ$, we can use the unit circle properties.

The coordinates $(x, y)$ of a point on the unit circle at an angle $\theta$ are given by:

$$x = \cos(\theta)$$

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

For $\theta = 45^\circ$:

$$x = \cos(45^\circ) = \frac{\sqrt{2}}{2}$$

$$y = \sin(45^\circ) = \frac{\sqrt{2}}{2}$$

So, the coordinates are:

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

Find the coordinates of a point on the unit circle, and determine the quadrant

Let the point on the unit circle have coordinates $(x, y)$, and let the angle it makes with the positive x-axis be $\theta = \frac{5\pi}{4}$ radians.

To find the coordinates:

$$x = \cos \frac{5\pi}{4}$$

$$y = \sin \frac{5\pi}{4}$$

Using the unit circle properties:

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

$$y = -\frac{\sqrt{2}}{2}$$

Since both coordinates are negative, the point lies in the third quadrant.

Determine the coordinates of the point on the unit circle corresponding to the angle 7π/6 radians

To find the coordinates of the point on the unit circle corresponding to the angle $\frac{7\pi}{6}$ radians, we need to consider the angle in standard position.

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

The reference angle is $\frac{\pi}{6}$ radians.

The coordinates for $\frac{\pi}{6}$ radians on the unit circle are $(\frac{\sqrt{3}}{2}, \frac{1}{2})$.

Since $\frac{7\pi}{6}$ is in the third quadrant, the coordinates are:

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

Find the cosine and sine of an angle on the unit circle

Given an angle $\theta = \frac{5\pi}{6}$ radians, find the coordinates $(\cos \theta, \sin \theta)$ on the unit circle.

Step 1: Recognize that $\theta = \frac{5\pi}{6}$ is an angle in the second quadrant.

Step 2: In the second quadrant, cosine is negative and sine is positive.

Step 3: Use the reference angle, which is $\pi – \frac{5\pi}{6} = \frac{\pi}{6}$.

Step 4: Recall the sine and cosine values for $\frac{\pi}{6}$: $\sin \frac{\pi}{6} = \frac{1}{2}$ and $\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$.

Step 5: Apply the signs for the second quadrant: $\cos \frac{5\pi}{6} = -\cos \frac{\pi}{6} = -\frac{\sqrt{3}}{2}$ and $\sin \frac{5\pi}{6} = \sin \frac{\pi}{6} = \frac{1}{2}$.

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

Find the value of cosecant for a given angle on the unit circle

Given that the angle \(\theta\) is \(\frac{5\pi}{6}\), find the value of \(csc(\theta)\) using the unit circle.

Step 1: First, locate the angle \(\frac{5\pi}{6}\) on the unit circle. This angle is in the second quadrant.

Step 2: The reference angle for \(\frac{5\pi}{6}\) is \(\frac{\pi}{6}\).

Step 3: The sine of \(\frac{5\pi}{6}\) is equal to the sine of \(\frac{\pi}{6}\) because they share the same reference angle.

Step 4: Thus, \(\sin(\frac{5\pi}{6}) = \sin(\frac{\pi}{6}) = \frac{1}{2}\).

Step 5: The cosecant function is the reciprocal of the sine function. Therefore, \(\csc(\frac{5\pi}{6}) = \frac{1}{\sin(\frac{5\pi}{6})} = \frac{1}{\frac{1}{2}} = 2\).