Math

Identify the Quadrant of an Angle in Radians

Given an angle of $ \frac{4\pi}{3} $ radians, determine the quadrant in which the terminal side of the angle lies.

First, recall that the unit circle is divided into four quadrants:

1. Quadrant I: $0 < \theta < \frac{\pi}{2}$

2. Quadrant II: $\frac{\pi}{2} < \theta < \pi$

3. Quadrant III: $\pi < \theta < \frac{3\pi}{2}$

4. Quadrant IV: $\frac{3\pi}{2} < \theta < 2\pi$

Here, $ \frac{4\pi}{3} $ radians is greater than $ \pi $ and less than $ \frac{3\pi}{2}$. Hence, it lies in Quadrant III.

Find the Cosine of an Angle on the Unit Circle

To find the cosine of an angle, we use the unit circle. Given that the angle is $\theta = \frac{\pi}{3}$, we need to find $\cos(\frac{\pi}{3})$.

On the unit circle, the coordinates of the point corresponding to the angle $\theta$ are $(\cos(\theta), \sin(\theta))$. For $\theta = \frac{\pi}{3}$, the coordinates are $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$. So,

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

Determine the tan values of specific angles on the unit circle

We need to determine the $\tan$ values for the angles $30^{\circ}$, $45^{\circ}$, and $60^{\circ}$ on the unit circle:

1. For $30^{\circ}$:

$$\tan 30^{\circ} = \frac{\sin 30^{\circ}}{\cos 30^{\circ}} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$$

2. For $45^{\circ}$:

$$\tan 45^{\circ} = \frac{\sin 45^{\circ}}{\cos 45^{\circ}} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$$

3. For $60^{\circ}$:

$$\tan 60^{\circ} = \frac{\sin 60^{\circ}}{\cos 60^{\circ}} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}$$

What are the sine, cosine, and tangent of the angle $\frac{\pi}{4}$ on the unit circle?

To find the sine, cosine, and tangent of the angle $\frac{\pi}{4}$ on the unit circle, we need to locate the angle on the circle.

The angle $\frac{\pi}{4}$ radians is equivalent to 45 degrees. In the unit circle, this corresponds to the point where both x and y coordinates are equal, as the angle bisects the first quadrant.

The coordinates of the point are $\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$.

Therefore:

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

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

$$ \tan \left( \frac{\pi}{4} \right) = \frac{ \frac{\sqrt{2}}{2} }{ \frac{\sqrt{2}}{2} } = 1 $$

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

To find $\csc(\frac{\pi}{3})$, we first need to recall the definition of the cosecant function:

$$\csc(\theta) = \frac{1}{\sin(\theta)}$$

Next, we locate the angle $\frac{\pi}{3}$ on the unit circle. The sine of $\frac{\pi}{3}$ is given by:

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

Now, using the definition of cosecant:

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

Therefore, $\csc(\frac{\pi}{3}) = \frac{2\sqrt{3}}{3}$.

Finding the Tangent of Angles on the Unit Circle

To find the tangent of an angle θ on the unit circle, we use the definition of tangent in terms of sine and cosine: $$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$$

Consider the angle θ = 45 degrees. The coordinates on the unit circle are (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}). Therefore,

$$\tan(45°) = \frac{\sin(45°)}{\cos(45°)} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 $$

What are the sine and cosine values of an angle of π/6 on the unit circle?

The angle $\frac{\pi}{6}$ radians corresponds to 30 degrees.

On the unit circle, the coordinates of the point at this angle represent the cosine and sine values.

Therefore, for the angle $\frac{\pi}{6}$:

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

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

Convert an angle from degrees to radians using the unit circle

The formula to convert degrees to radians is: \( \theta = \frac{\pi}{180} \times \text{degrees} \)

Given an angle of 120 degrees, we use the following calculation:

$$ \theta = \frac{\pi}{180} \times 120 $$

This simplifies to:

$$ \theta = \frac{2\pi}{3} $$

Hence, 120 degrees is equivalent to \( \frac{2\pi}{3} \) radians.

Determine the exact values of cotangent for an angle that, when doubled, corresponds to a point on the unit circle where the y-coordinate is equal to the negative square root of 3 divided by 2

To find $\cot(\theta)$, we need to determine the appropriate angle $2\theta$. Given that the y-coordinate of $2\theta$ is $-\frac{\sqrt{3}}{2}$, we know that $2\theta$ corresponds to $240^\circ$ or $300^\circ$ in the unit circle.

1. For $2\theta = 240^\circ$:

$$\theta = \frac{240^\circ}{2} = 120^\circ$$

$$\cot(120^\circ) = \cot(180^\circ – 60^\circ) = -\cot(60^\circ) = -\frac{1}{\sqrt{3}}$$

2. For $2\theta = 300^\circ$:

$$\theta = \frac{300^\circ}{2} = 150^\circ$$

$$\cot(150^\circ) = \cot(180^\circ – 30^\circ) = -\cot(30^\circ) = -\sqrt{3}$$

Therefore, the exact values of $\cot(\theta)$ are $-\frac{1}{\sqrt{3}}$ and $-\sqrt{3}$.

Find the value of tan(4π/3) on the unit circle

To find $ \tan \left( \frac{4\pi}{3} \right) $ on the unit circle, we note that $ \frac{4\pi}{3} $ radians is in the third quadrant.

In the third quadrant, both sine and cosine are negative. The reference angle for $ \frac{4\pi}{3} $ is $ \frac{\pi}{3} $.

We know that:

$$ \tan \left( \frac{\pi}{3} \right) = \sqrt{3} $$

Since tangent is positive in the third quadrant:

$$ \tan \left( \frac{4\pi}{3} \right) = \tan \left( \pi + \frac{\pi}{3} \right) = \tan \left( \frac{\pi}{3} \right) = \sqrt{3} $$

Therefore, $ \tan \left( \frac{4\pi}{3} \right) = \sqrt{3} $