Unit Circle

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:

– $$0^\circ (1,0)$$

– $$90^\circ (0,1)$$

– $$180^\circ (-1,0)$$

– $$270^\circ (0,-1)$$

– $$360^\circ (1,0)$$

What is the value of tan(45°) on the unit circle?

To find the value of $\tan(45°)$ on the unit circle, we use the definition of tangent:

$$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$$

At $\theta = 45°$, we have:

$$\sin(45°) = \frac{\sqrt{2}}{2}$$

$$\cos(45°) = \frac{\sqrt{2}}{2}$$

Thus,

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

So, the value of $\tan(45°)$ is $1$.

Find all angles θ in radians such that tan(θ) = 3 and θ is in the interval [0, 2π]

To solve the problem, we need to find all angles $\theta$ such that $\tan(\theta) = 3$ within the interval $[0, 2\pi]$.

Step 1: Recognize that $\tan(\theta)$ is positive in the first and third quadrants.

Step 2: The reference angle $\alpha$ for $\tan(\alpha) = 3$ is found using $\alpha = \arctan(3)$.

Step 3: Calculate $\alpha$:
$\alpha = \arctan(3) \approx 1.249$ radians.

Step 4: Identify the angles in the first and third quadrants:
$\theta_1 = \alpha = \arctan(3) \approx 1.249$ radians
$\theta_2 = \pi + \alpha = \pi + \arctan(3) \approx 4.391$ radians.

Therefore, the solutions are $\theta \approx 1.249$ radians and $\theta \approx 4.391$ radians.

Find the values of sin(30°) and cos(30°) on the unit circle

To find the values of $\sin(30°)$ and $\cos(30°)$ on the unit circle, we use the fact that 30° corresponds to $\frac{\pi}{6}$ radians.

The coordinates of the point on the unit circle at an angle $\frac{\pi}{6}$ from the positive x-axis are $(\cos(\frac{\pi}{6}), \sin(\frac{\pi}{6}))$.

We know:
$\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}$
$\sin(\frac{\pi}{6}) = \frac{1}{2}$

Therefore, $\sin(30°) = \frac{1}{2}$ and $\cos(30°) = \frac{\sqrt{3}}{2}$.

What is the value of cotangent at an angle of 45 degrees on the unit circle?

To find the cotangent of 45 degrees, we use the definition of cotangent on the unit circle:

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

For $\theta = 45^\circ$, we know that:

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

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

Therefore,

$$\cot(45^\circ) = \frac{\cos(45^\circ)}{\sin(45^\circ)} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$$

So, the value of $\cot(45^\circ)$ is 1.