Unit Circle

Find the Sine and Cosine of an Angle Given in a Flipped Unit Circle Problem

Given an angle $\theta$ in the flipped unit circle, where the x-values represent the sine of the angle and the y-values represent the cosine of the angle, find the sine and cosine of $\theta = \frac{5\pi}{4}$.

First, note that $\theta = \frac{5\pi}{4}$ is in the third quadrant. In the standard unit circle, $\sin(\frac{5\pi}{4}) = -\frac{1}{\sqrt{2}}$ and $\cos(\frac{5\pi}{4}) = -\frac{1}{\sqrt{2}}$.

Since the roles of sine and cosine are flipped, the sine of $\theta$ will be the x-coordinate, and the cosine of $\theta$ will be the y-coordinate.

Hence, the sine of $\theta = \frac{5\pi}{4}$ is $-\frac{1}{\sqrt{2}}$, and the cosine of $\theta = \frac{5\pi}{4}$ is $-\frac{1}{\sqrt{2}}$.

Find the value of sin(45°) using the unit circle

First, we need to locate the angle 45° on the unit circle. The coordinates of this angle on the unit circle are (√2/2, √2/2).

The sine of the angle is the y-coordinate of the point on the unit circle corresponding to that angle.

Therefore,

$$\sin(45°) = \frac{\sqrt{2}}{2}$$

Find the exact values of cotangent for specific angles on the unit circle

To find the exact values of $\cot$ for specific angles on the unit circle, let’s consider the angle $\theta = \frac{11\pi}{6}$.

Step 1: Identify the coordinates on the unit circle: The angle $\theta = \frac{11\pi}{6}$ corresponds to the point $\left( \cos(\frac{11\pi}{6}), \sin(\frac{11\pi}{6}) \right)$.

Step 2: Use the coordinates to find the cotangent: We know $\cos(\frac{11\pi}{6}) = \frac{\sqrt{3}}{2}$ and $\sin(\frac{11\pi}{6}) = -\frac{1}{2}$.

Step 3: Calculate $\cot(\theta)$: Using $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$, we get

$$ \cot\left( \frac{11\pi}{6} \right) = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3} $$

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

To find the value of $\cos(-\pi/3)$ on the unit circle, we should first recall the basic properties of the cosine function and the unit circle:

1. The cosine function is an even function, meaning $\cos(-x) = \cos(x)$.

2. Therefore, $\cos(-\pi/3) = \cos(\pi/3)$.

3. We know from the unit circle that $\cos(\pi/3) = \frac{1}{2}$.

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

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

Find the sine of the angle θ if θ is π/6 radians on the unit circle

To find the sine of the angle $\theta$ when $\theta = \frac{\pi}{6}$ radians:

Step 1: Locate $\frac{\pi}{6}$ on the unit circle. The angle $\frac{\pi}{6}$ is 30 degrees.

Step 2: Use the definition of sine on the unit circle, which is the y-coordinate of the point where the terminal side of the angle intersects the unit circle.

Step 3: For $\theta = \frac{\pi}{6}$, the coordinates on the unit circle are $(\frac{\sqrt{3}}{2}, \frac{1}{2})$.

Therefore, $\sin(\frac{\pi}{6}) = \frac{1}{2}$.

Finding the Sine of an Angle Using the Unit Circle

Given a point on the unit circle corresponding to an angle of \( \frac{\pi}{6} \) (30°), determine the sine of the angle.

The unit circle has a radius of 1. For an angle of \( \frac{\pi}{6} \), the coordinates are:

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

Therefore, the sine of \( \frac{\pi}{6} \) is:

$$ \sin \frac{\pi}{6} = \frac{1}{2} $$

Find the coordinates of the point on the unit circle corresponding to an angle of $\frac{\pi}{3}$ radians

To find the coordinates of the point on the unit circle at an angle of $\frac{\pi}{3}$ radians, we use the cosine and sine functions.

The coordinates are given by: $$ (\cos(\frac{\pi}{3}), \sin(\frac{\pi}{3})) $$

First, calculate the cosine: $$ \cos(\frac{\pi}{3}) = \frac{1}{2} $$

Next, calculate the sine: $$ \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} $$

Therefore, the coordinates are: $$ \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) $$

Find all angles θ in radians on the unit circle where sin(θ) = 1/2 and cos(θ) = -√3/2

To find angles θ where $\sin(\theta) = \frac{1}{2}$ and $\cos(\theta) = -\frac{\sqrt{3}}{2}$, we start by identifying possible angles for each trigonometric condition separately:

From $\sin(\theta) = \frac{1}{2}$, the possible angles are $\theta = \frac{\pi}{6}$ and $\theta = \frac{5\pi}{6}$ in the first and second quadrants.

From $\cos(\theta) = -\frac{\sqrt{3}}{2}$, the possible angles are $\theta = \frac{5\pi}{6}$ and $\theta = \frac{7\pi}{6}$ in the second and third quadrants.

The common angle satisfying both conditions is $\theta = \frac{5\pi}{6}$. Therefore, the solution is:

$$ \boxed{\frac{5\pi}{6}} $$

Find the equation of a unit circle

To find the equation of a unit circle centered at the origin, we need to remember that a unit circle has a radius of 1. The standard form of a circle’s equation is:

$$ (x – h)^2 + (y – k)^2 = r^2 $$

Where (h, k) is the center of the circle and r is the radius. Since the unit circle is centered at the origin (0, 0) and has a radius of 1, we can plug in these values:

$$ (x – 0)^2 + (y – 0)^2 = 1^2 $$

Simplifying this, we get:

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

The equation of the unit circle is:

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

Find the secant value of angle π/3 on the unit circle

To find the secant value, we first need to know the cosine value of the given angle on the unit circle.

The angle $\frac{\pi}{3}$ corresponds to an angle of $60^\circ$.

On the unit circle, the cosine of $\frac{\pi}{3}$ is $\frac{1}{2}$.

The secant is the reciprocal of the cosine:

$$\sec\left(\frac{\pi}{3}\right) = \frac{1}{\cos\left(\frac{\pi}{3}\right)} = \frac{1}{\frac{1}{2}} = 2$$