Math

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.

Find the sine and cosine values for the angle 5π/6

To find the sine and cosine values for the angle $\frac{5\pi}{6}$, we first understand that this angle is located in the second quadrant of the unit circle.

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

We know the sine and cosine values for $\frac{\pi}{6}$ are $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$ and $\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$.

Since $\frac{5\pi}{6}$ is in the second quadrant, the sine value remains positive, and the cosine value becomes negative.

Therefore, $\sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}$ and $\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}$.

Find the angles on the unit circle where the cosine value is equal to -1/2

To find the angles on the unit circle where the cosine value is equal to $-\frac{1}{2}$, we start by considering the unit circle properties and the cosine function.

The cosine value $-\frac{1}{2}$ corresponds to specific angles whose coordinates on the unit circle have an x-value of $-\frac{1}{2}$. These angles are found in the second and third quadrants of the unit circle.

We first identify the reference angle associated with the cosine value of $\frac{1}{2}$, which is $60^\circ$ or $\frac{\pi}{3}$ radians. Hence, the angles where the cosine is $-\frac{1}{2}$ are as follows:

1. Second quadrant: $180^\circ – 60^\circ = 120^\circ$ or $\pi – \frac{\pi}{3} = \frac{2\pi}{3}$ radians.

2. Third quadrant: $180^\circ + 60^\circ = 240^\circ$ or $\pi + \frac{\pi}{3} = \frac{4\pi}{3}$ radians.

Therefore, the angles on the unit circle where the cosine value is $-\frac{1}{2}$ are $120^\circ$ and $240^\circ$ or $\frac{2\pi}{3}$ and $\frac{4\pi}{3}$ radians.

Determine the coordinates of the point where the terminal side of an angle θ = 5π/4 radians intersects the unit circle

To find the coordinates of the point where the terminal side of an angle $\theta = \frac{5\pi}{4}$ intersects the unit circle, we start by expressing the angle in degrees:

$$\theta = \frac{5\pi}{4} \cdot \frac{180}{\pi} = 225^{\circ}$$

This angle is in the third quadrant where both sine and cosine are negative. For the unit circle, we can use the reference angle:

$$ 225^{\circ} – 180^{\circ} = 45^{\circ} $$

The coordinates corresponding to $45^{\circ}$ are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.

Since $225^{\circ}$ is in the third quadrant:

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

Therefore, the coordinates are:

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

Find the value of cos(θ) and sin(θ) for θ = 225°

First, we need to find the reference angle for $\theta = 225^\circ$. Since $225^\circ$ is in the third quadrant, the reference angle is:

$$225^\circ – 180^\circ = 45^\circ$$

In the third quadrant, the cosine and sine values are negative. For a $45^\circ$ reference angle, we have:

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

Thus, in the third quadrant:

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

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

Find the value of cos(-π/3) using the unit circle

To find $\cos(-\pi / 3)$, we can start by recognizing that the cosine function is even. This means $\cos(-x) = \cos(x)$. Therefore:

$$\cos(-\pi / 3) = \cos(\pi / 3)$$

From the unit circle, we know that:

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

So, the value of $\cos(-\pi / 3)$ is:

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

Find the equation of the unit circle centered at the origin

To find the equation of the unit circle centered at the origin, we start with the standard form of the circle equation:

$$ (x – h)^2 + (y – k)^2 = r^2 $$

For a unit circle centered at the origin, the center (h, k) is (0, 0) and the radius r is 1. Substituting these values, we get:

$$ (x – 0)^2 + (y – 0)^2 = 1^2 $$

Simplifying this, the equation of the unit circle is:

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