Homework

Techniques to Quickly Memorize the Unit Circle

$$ \text{One effective method is to use mnemonic devices and repetition.} $$

$$ \text{For instance, you can remember the coordinates for special angles, like } \theta = 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \text{ and so on.} $$

$$ \text{By repeating these values and using flashcards, you can reinforce memory.} $$

$$ \text{Additionally, understanding the symmetry of the unit circle can help. For example, the coordinates of } \frac{\pi}{6} \text{ (which are } (\frac{\sqrt{3}}{2}, \frac{1}{2})) \text{ can be reflected across the axes to find the coordinates for other angles like } \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}. $$

$$ \text{This approach takes advantage of patterns and reduces the amount of raw data you need to memorize.} $$

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:

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