Math

Calculate cos(-π/3) on the unit circle

To find $\cos(-\pi/3)$, we first need to understand its position on the unit circle. The angle $-\pi/3$ is equivalent to rotating $\pi/3$ radians in the clockwise direction.

On the unit circle, $\pi/3$ radians is located in the first quadrant, and its coordinates are $(1/2, \sqrt{3}/2)$. Since we are rotating clockwise, we need to reflect over the x-axis, thus the coordinates become $(1/2, -\sqrt{3}/2)$.

Therefore, $\cos(-\pi/3) = \cos(\pi/3) = 1/2$.

So, $$\cos(-\pi/3) = 1/2$$

Calculate the value of tan(4π/3) using the unit circle

To calculate $\tan\left(\frac{4\pi}{3}\right)$, we start by locating the angle $\frac{4\pi}{3}$ on the unit circle.

The angle $\frac{4\pi}{3}$ radians is equivalent to $240^\circ$.

This angle lies in the third quadrant where both sine and cosine are negative.

Using the unit circle, we find the coordinates of the point at $240^\circ$: $(\cos 240^\circ, \sin 240^\circ) = \left( -\frac{1}{2}, -\frac{\sqrt{3}}{2} \right)$.

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

$$\tan\left(\frac{4\pi}{3}\right) = \frac{\sin 240^\circ}{\cos 240^\circ} = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \sqrt{3}.$$

Determine the Quadrant of a Trigonometric Function

Consider the angle $\theta = 210^\circ$. To determine the quadrant where this angle lies, we will use the unit circle.

The angle $210^\circ$ is measured from the positive x-axis in the counter-clockwise direction. Since $210^\circ > 180^\circ$ and $210^\circ < 270^\circ$, it lies in the third quadrant.

In the third quadrant, both sine and cosine values are negative. Therefore, the angle $\theta = 210^\circ$ lies in Quadrant III.

Find the coordinates of the point on the unit circle corresponding to 210 degrees

The unit circle is a circle with radius 1 centered at the origin (0,0) in the coordinate plane. Points on the unit circle can be represented as (cos θ, sin θ), where θ is the angle in degrees.

To find the coordinates of the point on the unit circle corresponding to $210^{\circ}$:

1. Convert the angle to radians: $210^{\circ} = \frac{210 \pi}{180} = \frac{7 \pi}{6}$.

2. Use the unit circle values:

$$\cos \left(\frac{7 \pi}{6} \right) = -\frac{\sqrt{3}}{2}$$

$$\sin \left(\frac{7 \pi}{6} \right) = -\frac{1}{2}$$

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

Calculate cos(-π / 3) using the unit circle

Using the unit circle, we know that the angle $-\pi / 3$ corresponds to moving $\pi / 3$ radians clockwise from the positive x-axis.

Since $\cos$ is the x-coordinate of the point on the unit circle, and moving $\pi / 3$ radians clockwise is the same as moving $2\pi – \pi / 3 = 5\pi / 3$ radians counterclockwise from the positive x-axis, we need to find the cosine of $5\pi / 3$.

On the unit circle, the coordinates of the angle $5\pi / 3$ are $(\frac{1}{2}, -\frac{\sqrt{3}}{2})$. Therefore, the value of $\cos(5\pi / 3)$, corresponding to $\cos(-\pi / 3)$, is $\frac{1}{2}$.

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

On the unit circle, find the values of angles in radians for which the secant function is undefined

The secant function $\sec(\theta)$ is undefined when the cosine function $\cos(\theta)$ is zero. On the unit circle, $\cos(\theta)$ is zero at the points where the x-coordinate is zero, which happens at $\theta = \frac{\pi}{2} + k\pi$ for any integer $k$.

Therefore, the angles in radians for which the secant function is undefined are:

$$\theta = \frac{\pi}{2} + k\pi$$

where $k \in \mathbb{Z}$ (any integer).

Find the value of \( \cot(\theta) \) when \( \theta = \frac{\pi}{4} \) on the unit circle

Given:

\( \theta = \frac{\pi}{4} \)

On the unit circle, the coordinates for \( \theta = \frac{\pi}{4} \) are:

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

\( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \)

Substituting the values:

\( \cot(\frac{\pi}{4}) = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 \)

Therefore, \( \cot(\frac{\pi}{4}) = 1 \).

Find the tangent of 45 degrees using the unit circle

To find the tangent of 45 degrees using the unit circle, we first locate the point corresponding to 45 degrees on the circle. The coordinates of this point are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.

Recall that the tangent function is defined as the ratio of the y-coordinate to the x-coordinate:

$$\tan(45^\circ) = \frac{\text{y-coordinate}}{\text{x-coordinate}} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$$

Therefore, $$\tan(45^\circ) = 1$$.

Find the coordinates of points on the unit circle corresponding to a given angle in a flipped configuration

Given a unit circle, we need to find the coordinates of points corresponding to the angle $\theta = \frac{5\pi}{4}$, but with the configuration flipped over the x-axis.

In the standard unit circle, the point corresponding to $\theta = \frac{5\pi}{4}$ is:

$$\left(\cos \frac{5\pi}{4}, \sin \frac{5\pi}{4}\right)$$

Using the trigonometric values:

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

Since the configuration is flipped over the x-axis, we change the sign of the y-coordinate:

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

Thus, the coordinates are:

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

Find the sine and cosine of the angle \(\theta\) when \(\theta = \frac{\pi}{6}\)

To find the sine and cosine of the angle $\theta$ when $\theta = \frac{\pi}{6}$, we can use the unit circle.

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

On the unit circle, the coordinates of the point at angle $\frac{\pi}{6}$ are:

$$\left(\cos\left(\frac{\pi}{6}\right), \sin\left(\frac{\pi}{6}\right)\right)$$

From trigonometric values:

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

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

Therefore, the sine and cosine of the angle $\theta$ when $\theta = \frac{\pi}{6}$ are given by:

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

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