Math

Finding Reference Angle for Angles not on the Unit Circle

To find the reference angle for an angle not on the unit circle, we first need to understand the definition of a reference angle. A reference angle is the acute angle formed by the terminal side of the given angle and the horizontal axis. The reference angle is always between 0° and 90°, and it is always positive.

Let’s consider an example: Find the reference angle for the angle 250°.

Step 1: Determine the quadrant in which the given angle lies. Since 250° is between 180° and 270°, it lies in the third quadrant.

Step 2: Use the formula for the reference angle in the third quadrant:

$$ \theta_{reference} = \theta – 180° $$

For our example:

$$ \theta_{reference} = 250° – 180° = 70° $$

Therefore, the reference angle for 250° is 70°.

What are the sine, cosine, and tangent of the angle π/3 on the unit circle?

First, locate the angle $\frac{\pi}{3}$ on the unit circle. This angle corresponds to 60 degrees.

The coordinates of the point on the unit circle at this angle are $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.

Thus, the cosine of $\frac{\pi}{3}$ is the x-coordinate, which is $\frac{1}{2}$:

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

The sine of $\frac{\pi}{3}$ is the y-coordinate, which is $\frac{\sqrt{3}}{2}$:

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

The tangent is given by the ratio of the sine to the cosine:

$$\tan \frac{\pi}{3} = \frac{\sin \frac{\pi}{3}}{\cos \frac{\pi}{3}} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}$$

What are the coordinates of the point on the unit circle at an angle of π/3 radians?

Given an angle of $\frac{\pi}{3}$ radians, we want to find the coordinates of the corresponding point on the unit circle.

The unit circle has a radius of 1, and the coordinates of any point on the unit circle can be found using the cosine and sine of the angle.

Therefore, the x-coordinate is $\cos(\frac{\pi}{3})$ and the y-coordinate is $\sin(\frac{\pi}{3})$.

We know from trigonometric values:

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

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

Thus, the coordinates are:

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

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.