Unit Circle

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

To find the sine and cosine values for the angle $\frac{5\pi}{6}$, we first understand that this angle is located in the second quadrant of the unit circle.

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

We know the sine and cosine values for $\frac{\pi}{6}$ are $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$ and $\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$.

Since $\frac{5\pi}{6}$ is in the second quadrant, the sine value remains positive, and the cosine value becomes negative.

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

Find the angles on the unit circle where the cosine value is equal to -1/2

To find the angles on the unit circle where the cosine value is equal to $-\frac{1}{2}$, we start by considering the unit circle properties and the cosine function.

The cosine value $-\frac{1}{2}$ corresponds to specific angles whose coordinates on the unit circle have an x-value of $-\frac{1}{2}$. These angles are found in the second and third quadrants of the unit circle.

We first identify the reference angle associated with the cosine value of $\frac{1}{2}$, which is $60^\circ$ or $\frac{\pi}{3}$ radians. Hence, the angles where the cosine is $-\frac{1}{2}$ are as follows:

1. Second quadrant: $180^\circ – 60^\circ = 120^\circ$ or $\pi – \frac{\pi}{3} = \frac{2\pi}{3}$ radians.

2. Third quadrant: $180^\circ + 60^\circ = 240^\circ$ or $\pi + \frac{\pi}{3} = \frac{4\pi}{3}$ radians.

Therefore, the angles on the unit circle where the cosine value is $-\frac{1}{2}$ are $120^\circ$ and $240^\circ$ or $\frac{2\pi}{3}$ and $\frac{4\pi}{3}$ radians.

Determine the coordinates of the point where the terminal side of an angle θ = 5π/4 radians intersects the unit circle

To find the coordinates of the point where the terminal side of an angle $\theta = \frac{5\pi}{4}$ intersects the unit circle, we start by expressing the angle in degrees:

$$\theta = \frac{5\pi}{4} \cdot \frac{180}{\pi} = 225^{\circ}$$

This angle is in the third quadrant where both sine and cosine are negative. For the unit circle, we can use the reference angle:

$$ 225^{\circ} – 180^{\circ} = 45^{\circ} $$

The coordinates corresponding to $45^{\circ}$ are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.

Since $225^{\circ}$ is in the third quadrant:

$$ \cos(225^{\circ}) = -\frac{\sqrt{2}}{2}, \sin(225^{\circ}) = -\frac{\sqrt{2}}{2} $$

Therefore, the coordinates are:

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

Find the value of cos(θ) and sin(θ) for θ = 225°

First, we need to find the reference angle for $\theta = 225^\circ$. Since $225^\circ$ is in the third quadrant, the reference angle is:

$$225^\circ – 180^\circ = 45^\circ$$

In the third quadrant, the cosine and sine values are negative. For a $45^\circ$ reference angle, we have:

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

Thus, in the third quadrant:

$$\cos(225^\circ) = -\frac{\sqrt{2}}{2}$$

$$\sin(225^\circ) = -\frac{\sqrt{2}}{2}$$

Find the value of cos(-π/3) using the unit circle

To find $\cos(-\pi / 3)$, we can start by recognizing that the cosine function is even. This means $\cos(-x) = \cos(x)$. Therefore:

$$\cos(-\pi / 3) = \cos(\pi / 3)$$

From the unit circle, we know that:

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

So, the value of $\cos(-\pi / 3)$ is:

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

Find the equation of the unit circle centered at the origin

To find the equation of the unit circle centered at the origin, we start with the standard form of the circle equation:

$$ (x – h)^2 + (y – k)^2 = r^2 $$

For a unit circle centered at the origin, the center (h, k) is (0, 0) and the radius r is 1. Substituting these values, we get:

$$ (x – 0)^2 + (y – 0)^2 = 1^2 $$

Simplifying this, the equation of the unit circle is:

$$ x^2 + y^2 = 1 $$

Determine the angle measures of a point on the unit circle

Given a point $P$ on the unit circle with coordinates $P = (\frac{3}{5}, -\frac{4}{5})$, determine all possible angle measures $\theta$ in degrees.

First, we calculate the reference angle $\alpha$ by using the trigonometric functions. Notice that the coordinates of $P$ give us the cosine and sine of $\theta$:

$$\cos(\theta) = \frac{3}{5}, \sin(\theta) = -\frac{4}{5}$$

Using the inverse cosine function, we find the reference angle:

$$\alpha = \cos^{-1}(\frac{3}{5}) \approx 53.13^\circ$$

Since the sine is negative and the cosine is positive, $\theta$ is in the fourth quadrant. Therefore,

$$ \theta = 360^\circ – \alpha = 360^\circ – 53.13^\circ \approx 306.87^\circ$$

Thus, the possible angle measures are:

$$ \theta \approx 306.87^\circ $$

Finding Specific Trigonometric Values on the Unit Circle

Given the angle θ = 225°, find the sine, cosine, and tangent values using the unit circle.

First, convert the angle 225° to radians: $$225° = \frac{5\pi}{4}$$ radians.

On the unit circle, the angle $$\frac{5\pi}{4}$$ is located in the third quadrant, where sine and cosine are both negative.

The coordinates for $$\frac{5\pi}{4}$$ are $$(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$$.

Thus, $$\sin(225°) = -\frac{\sqrt{2}}{2}$$, $$\cos(225°) = -\frac{\sqrt{2}}{2}$$

Finally, the tangent value: $$\tan(225°) = \frac{\sin(225°)}{\cos(225°)} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1$$.

Therefore, $$\sin(225°) = -\frac{\sqrt{2}}{2}$$, $$\cos(225°) = -\frac{\sqrt{2}}{2}$$, $$\tan(225°) = 1$$.

Find the tangent of the angle

Given an angle \( \theta = \frac{\pi}{4} \), find \( \tan(\theta) \).

Since the angle \( \theta \) is within the first quadrant and \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \), we have:

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

Therefore:

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

So, \( \tan(\frac{\pi}{4}) = 1 \).

Find the cotangent of the angle $\theta = \frac{\pi}{4}$ on the unit circle

To find the cotangent of $\theta = \frac{\pi}{4}$ on the unit circle:

The cotangent function is given by:

$$\cot \theta = \frac{1}{\tan \theta}$$

Since $\tan \theta = \frac{\sin \theta}{\cos \theta}$, we first find the values of $\sin \theta$ and $\cos \theta$. For $\theta = \frac{\pi}{4}$:

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

Then:

$$\tan \frac{\pi}{4} = \frac{\sin \frac{\pi}{4}}{\cos \frac{\pi}{4}} = 1$$

Thus, the cotangent is:

$$\cot \frac{\pi}{4} = \frac{1}{\tan \frac{\pi}{4}} = 1$$

So, the answer is:

$$\cot \frac{\pi}{4} = 1$$