Math

Find the coordinates of the point where the angle 5π/4 radians intersects the unit circle

First, we need to convert the angle $\frac{5\pi}{4}$ radians into degrees. We know that:

$$ 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} $$

Thus,

$$ \frac{5\pi}{4} \text{ radians} = \frac{5\pi}{4} \times \frac{180}{\pi} \text{ degrees} = 225 \text{ degrees} $$

On the unit circle, the coordinates corresponding to an angle of $225^{\circ}$ (or $\frac{5\pi}{4}$ radians) can be found using the cosine and sine functions:

$$ \cos(225^{\circ}) = -\frac{\sqrt{2}}{2} $$

$$ \sin(225^{\circ}) = -\frac{\sqrt{2}}{2} $$

Therefore, the coordinates are:

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

Locate -π/2 on a Unit Circle

To locate $-\pi/2$ on the unit circle, we can follow these steps:

1. Start at the positive x-axis (0 radians).

2. Move clockwise because the angle is negative.

3. Since $-\pi/2$ radians equals -90 degrees, move 90 degrees clockwise from the positive x-axis.

4. This will place you on the negative y-axis.

Therefore, the coordinates for $-\pi/2$ on the unit circle are (0, -1).

Find the Trigonometric Values Using the Unit Circle

Given the angle $\theta = \frac{2\pi}{3}$, find the values of $\sin(\theta)$, $\cos(\theta)$, and $\tan(\theta)$ using the unit circle on a GDC TI calculator.

1. Locate the angle $\theta = \frac{2\pi}{3}$ on the unit circle.

2. The coordinates of the point where the terminal side intersects the unit circle are $(-\frac{1}{2}, \frac{\sqrt{3}}{2})$.

3. Hence, $\sin(\theta) = \frac{\sqrt{3}}{2}$, $\cos(\theta) = -\frac{1}{2}$, and $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\sqrt{3}}{2} \div -\frac{1}{2} = -\sqrt{3}$.

Therefore, the trigonometric values are:

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

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

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

Find the exact values of the trigonometric functions at an angle of 7π/6 on the unit circle

To solve this problem, let’s first locate the angle $\frac{7\pi}{6}$ on the unit circle. This angle is in the third quadrant because $\frac{7\pi}{6}$ is greater than $\pi$ but less than $\frac{3\pi}{2}$.

The reference angle is calculated by subtracting $\pi$ from $\frac{7\pi}{6}$:

$$\frac{7\pi}{6} – \pi = \frac{7\pi}{6} – \frac{6\pi}{6} = \frac{\pi}{6}$$

In the third quadrant, both sine and cosine are negative. The reference angle $\frac{\pi}{6}$ has sine and cosine values of $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$ respectively.

Therefore, the exact values of the trigonometric functions at $\frac{7\pi}{6}$ are:

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

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

$$ \tan \left( \frac{7\pi}{6} \right) = \frac{\sin \left( \frac{7\pi}{6} \right)}{\cos \left( \frac{7\pi}{6} \right)} = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} $$

Suppose you have a unit circle centered at the origin in the coordinate plane You flip the unit circle over the y-axis Determine the coordinates of a point (x, y) on the original unit circle after the transformation, given that x^2 + y^2 = 1

$$\text{Given the equation of the original unit circle}$$

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

$$\text{When the unit circle is flipped over the y-axis, each point } (x, y) \text{ is transformed to } (-x, y).$$

$$\text{So, the new coordinates after transformation are } (-x, y).$$

$$\text{For instance, if you have a point } (x, y) = (\frac{1}{2}, \frac{\sqrt{3}}{2}) \text{ on the original unit circle, the transformed coordinates are:}$$

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

Find the exact values of sine and cosine for an angle of 5π/4 radians on the unit circle

To solve for sine and cosine of the angle $\frac{5\pi}{4}$, we first determine its location on the unit circle.

The angle $\frac{5\pi}{4}$ radians is in the third quadrant, where both sine and cosine values are negative.

The reference angle for $\frac{5\pi}{4}$ radians is $\pi/4$ radians, whose sine and cosine values are $\frac{\sqrt{2}}{2}$ and $\frac{\sqrt{2}}{2}$ respectively.

Thus, for $\frac{5\pi}{4}$:

$$\sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$$

$$\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$$

Find the value of sin(30 degrees) on the unit circle

To find the value of $\sin(30^\circ)$ on the unit circle, we first need to recognize that $30^\circ$ is a special angle. On the unit circle, the angle $30^\circ$ corresponds to the coordinates $(\frac{\sqrt{3}}{2}, \frac{1}{2})$. The sine function gives the y-coordinate of this point.

Therefore,

$$\sin(30^\circ) = \frac{1}{2}.$$

Identify the cosine and sine values of 45° using the unit circle

To find the cosine and sine values of 45° using the unit circle, we first recognize that 45° corresponds to the angle π/4 radians.

In the unit circle, the coordinates of the point where the terminal side of the angle intersects the circle provide the cosine and sine values.

At 45° (π/4), both the x-coordinate (cosine) and y-coordinate (sine) are equal. They are both equal to 1/√2, which simplifies to √2/2.

Therefore, for 45°:

$$\cos(45°) = \frac{\sqrt{2}}{2}$$

$$\sin(45°) = \frac{\sqrt{2}}{2}$$

Techniques to Remember the Unit Circle for High School Students

One way to remember the unit circle is by focusing on the key angles and their coordinates. Let’s start with the four quadrants: $$0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$$ radians or $$0^\circ, 90^\circ, 180^\circ, 270^\circ, 360^\circ$$. The coordinates for these angles are as follows:

– $$0^\circ (1,0)$$

– $$90^\circ (0,1)$$

– $$180^\circ (-1,0)$$

– $$270^\circ (0,-1)$$

– $$360^\circ (1,0)$$