Unit Circle

Find the sine and cosine values for \( \frac{\pi}{4} \) using the unit circle

To find the sine and cosine values for $\frac{\pi}{4}$ using the unit circle, we use the coordinates of the point corresponding to that angle on the circle.

The angle $\frac{\pi}{4}$ is equivalent to 45 degrees. On the unit circle, the coordinates of the point at this angle are $\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$.

Therefore:

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

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

Calculate the exact value of tan(-π/6) using the unit circle and verify by applying trigonometric identities

Using the unit circle, first note that $-\frac{\pi}{6}$ is equivalent to $-30^\circ$. On the unit circle, this angle corresponds to the coordinates $\left( \frac{\sqrt{3}}{2}, -\frac{1}{2} \right)$.

Therefore, the value of $\tan(-\frac{\pi}{6})$ is given by the ratio of the y-coordinate to the x-coordinate:

$$ \tan(-\frac{\pi}{6}) = \frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} $$

Verification using trigonometric identities can be done by noting that $\tan(-x) = -\tan(x)$. Hence,

$$ \tan(-\frac{\pi}{6}) = -\tan(\frac{\pi}{6}) = -\frac{\sqrt{3}}{3} $$

Find the exact values of sin(225°) and cos(225°) using the unit circle

To find the exact values of $\sin(225°)$ and $\cos(225°)$, we first locate $225°$ on the unit circle.

$225°$ is in the third quadrant. The reference angle is $225° – 180° = 45°$. In the third quadrant, the sine and cosine of the reference angle are both negative.

Therefore, $\sin(225°) = -\sin(45°) = – \frac{\sqrt{2}}{2}$ and $\cos(225°) = -\cos(45°) = -\frac{\sqrt{2}}{2}$.

Thus, the exact values are:

$$\sin(225°) = -\frac{\sqrt{2}}{2}$$

$$\cos(225°) = -\frac{\sqrt{2}}{2}$$

Find the coordinates of the point on the unit circle where the angle is π/10 radians

To find the coordinates of the point on the unit circle at an angle of $\frac{\pi}{10}$ radians, we use the cosine and sine functions:

$$x = \cos\left(\frac{\pi}{10}\right)$$

$$y = \sin\left(\frac{\pi}{10}\right)$$

Therefore, the coordinates are:

$$\left( \cos\left(\frac{\pi}{10}\right), \sin\left(\frac{\pi}{10}\right) \right)$$

Find all the solutions for cos(θ) = -1/2 on the unit circle

$$ \text{We need to find all } \theta \text{ such that } \cos(\theta) = -\frac{1}{2}. $$

$$ \text{The values of } \theta \text{ where } \cos(\theta) = -\frac{1}{2} \text{ are at } \theta = \frac{2\pi}{3} + 2k\pi \text{ and } \theta = \frac{4\pi}{3} + 2k\pi \text{ for any integer } k. $$

$$ \text{Thus, all solutions are: } \theta = \frac{2\pi}{3} + 2k\pi \text{ and } \theta = \frac{4\pi}{3} + 2k\pi. $$

Find the value of cotangent for an angle of 45 degrees on the unit circle

To find $\cot(45^\circ)$ on the unit circle, we start by recalling that cotangent is the reciprocal of the tangent function.

Given:

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

For $\theta = 45^\circ$:

$$\tan(45^\circ) = 1 $$

Thus,

$$\cot(45^\circ) = \frac{1}{1} = 1 $$

Hence, the value of $\cot(45^\circ)$ is 1.

Find the Quadrant of a Given Angle on the Unit Circle

To determine the quadrant in which the angle 150° lies, we first convert it to radians:

$$150° \times \frac{\pi}{180°} = \frac{5\pi}{6}$$

The angle \(\frac{5\pi}{6}\) is greater than \(\frac{\pi}{2}\) but less than \(\pi\). Hence, it lies in the second quadrant.

Find the sine and cosine of the angle at three specific points on the unit circle

The three specific points we will consider are $\frac{\pi}{6}$, $\frac{\pi}{4}$, and $\frac{\pi}{3}$.

1. For $\frac{\pi}{6}$:

The sine value can be found using the unit circle as $\sin(\frac{\pi}{6}) = \frac{1}{2}$.

The cosine value can be found as $\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}$.

2. For $\frac{\pi}{4}$:

The sine value can be found as $\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$.

The cosine value can be found as $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$.

3. For $\frac{\pi}{3}$:

The sine value can be found as $\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$.

The cosine value can be found as $\cos(\frac{\pi}{3}) = \frac{1}{2}$.

Therefore, the values are:

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

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

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

Given a point on the unit circle at coordinates (sqrt(3)/2, -1/2), find the angle in radians, the corresponding angle in degrees, and the sine of the angle

To find the angle in radians, we use the coordinates on the unit circle. The x-coordinate gives us $$\cos(\theta) = \frac{\sqrt{3}}{2}$$ and the y-coordinate gives us $$\sin(\theta) = -\frac{1}{2}$$. These coordinates correspond to an angle in the fourth quadrant. Therefore, the angle in radians is:

$$\theta = -\frac{\pi}{6}$$

To convert this to degrees, we use the conversion factor $$\frac{180}{\pi}$$:

$$\theta = -\frac{\pi}{6} \times \frac{180}{\pi} = -30^\circ$$

The sine of the angle is already given by the y-coordinate:

$$\sin(\theta) = -\frac{1}{2}$$

Find the sine, cosine, and tangent of π/6 on the unit circle

To find the sine, cosine, and tangent of $\frac{\pi}{6}$, we use the unit circle values:

Sine of $\frac{\pi}{6}$: $$\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$$

Cosine of $\frac{\pi}{6}$: $$\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$$

Tangent of $\frac{\pi}{6}$: $$\tan\left(\frac{\pi}{6}\right) = \frac{\sin\left(\frac{\pi}{6}\right)}{\cos\left(\frac{\pi}{6}\right)} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$$