Unit Circle

Determine the Location of -π/2 on a Unit Circle

To determine the location of $-\pi/2$ on a unit circle, we follow these steps:

1. Understand that the unit circle is a circle with a radius of 1 centered at the origin (0,0) in the coordinate plane.

2. The angle $-\pi/2$ is measured in radians and indicates a rotation of 90 degrees in the clockwise direction from the positive x-axis.

3. On the unit circle, $-\pi/2$ radians corresponds to the point where the angle terminates. Moving 90 degrees clockwise from the positive x-axis places the terminal side of the angle along the negative y-axis.

Therefore, the coordinates of the point corresponding to $-\pi/2$ are:

$$(-\pi/2) = (0, -1)$$

Thus, the point on the unit circle corresponding to the angle $-\pi/2$ is (0, -1).

Find the sine, cosine, and tangent values for the angle $\theta = \frac{5\pi}{6}$ using the unit circle

For the angle $\theta = \frac{5\pi}{6}$:

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

In the second quadrant, sine is positive, cosine is negative, and tangent is negative.

Thus, the values are:

$\sin(\frac{5\pi}{6}) = \sin(\frac{\pi}{6}) = \frac{1}{2}$

$\cos(\frac{5\pi}{6}) = -\cos(\frac{\pi}{6}) = -\frac{\sqrt{3}}{2}$

$\tan(\frac{5\pi}{6}) = \frac{\sin(\frac{5\pi}{6})}{\cos(\frac{5\pi}{6})} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$

Ways to Memorize the Unit Circle

$$Ways to Memorize the Unit Circle$$

Explanation with Examples:

The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the coordinate plane. To memorize the unit circle, follow these steps.

1. Know the Key Angles:

Memorize the common angles in radians: 0, $$\frac{\pi}{6}$$, $$\frac{\pi}{4}$$, $$\frac{\pi}{3}$$, $$\frac{\pi}{2}$$, $$\pi$$, $$\frac{3\pi}{2}$$, and $$2\pi$$.

2. Memorize the Coordinates:

For each angle, memorize the coordinates on the unit circle.

For instance:

$$\text{At }\theta = 0\text{ or }2\pi,$$

$$(cos(\theta), sin(\theta)) = (1, 0)$$

$$\text{At }\theta = \frac{\pi}{2},$$

$$(cos(\theta), sin(\theta)) = (0, 1)$$

$$\text{At }\theta = \pi,$$

$$(cos(\theta), sin(\theta)) = (-1, 0)$$

$$\text{At }\theta = \frac{3\pi}{2},$$

$$(cos(\theta), sin(\theta)) = (0, -1)$$

3. Use Mnemonics:

Use mnemonic devices to remember the coordinates, such as the phrase ‘All Students Take Calculus’ to remember the signs of the coordinates in each quadrant.

Find the coordinates of a point on the unit circle corresponding to a given angle

Given an angle of \( \theta = 45^{\circ} \). To find the coordinates of the point on the unit circle:

The coordinates of any point on the unit circle can be found using the formulas:

\[ x = \cos(\theta) \]

\[ y = \sin(\theta) \]

Using \( \theta = 45^{\circ} \):

\[ x = \cos(45^{\circ}) = \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \]

\[ y = \sin(45^{\circ}) = \sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \]

The coordinates are \( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \).

Find the value of tan(θ) on the unit circle where θ is π/4

On the unit circle, the coordinates for $\theta = \frac{\pi}{4}$ are $\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$.

Therefore, $\tan(\frac{\pi}{4})$ is calculated as:

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

If point P on the unit circle is flipped over the y-axis, what will be the coordinates of point P if it initially lies on the point (sqrt(3)/2, 1/2)?

Initial coordinates of point $P$ are $(\frac{\sqrt{3}}{2}, \frac{1}{2})$. When flipped over the $y$-axis, the x-coordinate becomes its negative value while the y-coordinate remains the same. Therefore, the new coordinates of point $P$ are:

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

Find the coordinates of a point on the unit circle at 45 degrees

The unit circle is a circle with a radius of 1 centered at the origin. To find the coordinates of a point at $45^{\circ}$, we use the trigonometric functions sine and cosine.

For $\theta = 45^{\circ}$:

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

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

Therefore, the coordinates of the point are:

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

Find the coordinates of the points on the unit circle where the angle formed with the positive x-axis is such that the cosine of the angle equals -3/5 Additionally, find the corresponding sine value

To solve this problem, we start with the unit circle equation:

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

Given that $\cos(\theta) = \frac{-3}{5}$, we know the x-coordinate is $\frac{-3}{5}$. Let’s find the y-coordinate (sine value).

Substituting $\cos(\theta)$ in the unit circle equation:

$$\left(\frac{-3}{5}\right)^2 + y^2 = 1$$

$$\frac{9}{25} + y^2 = 1$$

Solving for $y^2$:

$$y^2 = 1 – \frac{9}{25}$$

$$y^2 = \frac{25}{25} – \frac{9}{25}$$

$$y^2 = \frac{16}{25}$$

Thus, $y = \pm \frac{4}{5}$.

The coordinates on the unit circle are:

$$\left( \frac{-3}{5}, \frac{4}{5} \right) \text{ and } \left( \frac{-3}{5}, \frac{-4}{5} \right)$$

Hence, the coordinates are $\left( \frac{-3}{5}, \frac{4}{5} \right) \text{ and } \left( \frac{-3}{5}, \frac{-4}{5} \right)$, and the corresponding sine values are $\frac{4}{5}$ and $\frac{-4}{5}$.

Find the Cartesian coordinates of a point on the unit circle where the angle is 135 degrees

To find the Cartesian coordinates of a point on the unit circle where the angle is $135^{\circ}$, we use the unit circle equation:

$$x = \cos(135^{\circ})$$

$$y = \sin(135^{\circ})$$

First, we calculate the cosine and sine of $135^{\circ}$:

$$\cos(135^{\circ}) = -\frac{\sqrt{2}}{2}$$

$$\sin(135^{\circ}) = \frac{\sqrt{2}}{2}$$

So, the Cartesian coordinates are:

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

Find the exact value of sin(2θ) and cos(2θ) given the point on the unit circle and the quadrant

Given the point $ P = \left( -\frac{3}{5}, -\frac{4}{5} \right) $ on the unit circle, find the exact values of $ \sin(2\theta) $ and $ \cos(2\theta) $. The point $ P $ is in Quadrant III.

To find $ \theta $, we use the definitions of sine and cosine on the unit circle:

$$ \sin(\theta) = y = -\frac{4}{5} $$

$$ \cos(\theta) = x = -\frac{3}{5} $$

Using the double angle formulas:

$$ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) $$

$$ \cos(2\theta) = \cos^2(\theta) – \sin^2(\theta) $$

Substituting the values:

$$ \sin(2\theta) = 2 \left( -\frac{4}{5} \right) \left( -\frac{3}{5} \right) = 2 \times \frac{12}{25} = \frac{24}{25} $$

$$ \cos(2\theta) = \left( -\frac{3}{5} \right)^2 – \left( -\frac{4}{5} \right)^2 = \frac{9}{25} – \frac{16}{25} = -\frac{7}{25} $$