Unit Circle

Determine the value of sec(θ) given cos(θ) = -1/sqrt(2) and θ is in the third quadrant

Given that $\cos(\theta) = -\frac{1}{\sqrt{2}}$ and $\theta$ is in the third quadrant, we start by using the identity:

$$\sec(\theta) = \frac{1}{\cos(\theta)}$$

Substituting the given value:

$$\sec(\theta) = \frac{1}{-\frac{1}{\sqrt{2}}}$$

We simplify the fraction:

$$\sec(\theta) = -\sqrt{2}$$

Thus, the value of $\sec(\theta)$ is $-\sqrt{2}$.

Find the sine and cosine values at t = pi/4 on the unit circle

To find the sine and cosine values at $ t = \frac{\pi}{4} $ on the unit circle, we use the following values:

$$ \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} $$

$$ \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} $$

Thus, the sine and cosine values are:

$$ \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}, \quad \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} $$

Identify the coordinates on the unit circle for θ = π/4

To find the coordinates on the unit circle for $ \theta = \frac{\pi}{4} $, we use the unit circle properties.

For $ \theta = \frac{\pi}{4} $, the coordinates are given by:

$$ (\cos(\frac{\pi}{4}), \sin(\frac{\pi}{4})) $$

From trigonometric values, we know:

$$ \cos(\frac{\pi}{4}) = \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} $$

So the coordinates are:

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

Find the values of sin and cos at the given angle on the unit circle

To find the values of $\sin$ and $\cos$ at an angle of $\theta = \frac{\pi}{3}$ on the unit circle:

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

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

Find the angle in radians for a point on the unit circle where the x-coordinate is 1/2

To find the angle $\theta$ in radians for a point on the unit circle where the $x$-coordinate is $\frac{1}{2}$, we consider the cosine function:

$$\cos(\theta) = \frac{1}{2}$$

The angles that satisfy this equation are:

$$\theta = \frac{\pi}{3}$$ and $$\theta = -\frac{\pi}{3}$$ or equivalently $$\theta = \frac{5\pi}{3}$$

What is the value of sin(15 degrees) using the unit circle?

To find the value of $ \sin(15^\circ) $ using the unit circle, we use the angle addition formula:

$$ \sin(a + b) = \sin(a) \cos(b) + \cos(a)\sin(b) $$

Here, let $ a = 45^\circ $ and $ b = -30^\circ $. Then,

$$ \sin(45^\circ – 30^\circ) = \sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ) $$

Substituting the values, we get:

$$ \sin(15^\circ) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} $$

Simplify the expression:

$$ \sin(15^\circ) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} $$

Combine the fractions:

$$ \sin(15^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4} $$

Solve for the trigonometric value of tan(x) when cos(x) = 1/2 on the unit circle

To solve for $ \tan(x) $ given $ \cos(x) = \frac{1}{2} $ on the unit circle, we must first determine the corresponding $ \sin(x) $. On the unit circle:

$$ \cos(x) = \frac{1}{2} $$

We know that at $ x = \frac{\pi}{3} $ and $ x = -\frac{\pi}{3} $, $ \cos(x) = \frac{1}{2} $. Correspondingly, $ \sin(x) $ at these points are:

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

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

Using the identity for tangent:

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

For $ x = \frac{\pi}{3} $:

$$ \tan(\frac{\pi}{3}) = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} $$

For $ x = -\frac{\pi}{3} $:

$$ \tan(-\frac{\pi}{3}) = \frac{ – \frac{\sqrt{3}}{2}}{\frac{1}{2}} = – \sqrt{3} $$

Find the coordinates of the point on the unit circle that corresponds to an angle of pi/3

To find the coordinates of the point on the unit circle at an angle of $ \pi/3 $ , we use the fact that the unit circle

Complete the unit circle with the correct trigonometric values

To complete the unit circle, fill in the following values:

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

$$\text{2. } \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}, \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$$

$$\text{3. } \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}, \cos(\frac{\pi}{3}) = \frac{1}{2}$$

Find the coordinates on the unit circle at an angle of π/3

To find the coordinates on the unit circle at an angle of $ \frac{\pi}{3} $, we use the unit circle definition:

$$ (\cos(\theta), \sin(\theta)) $$

For $ \theta = \frac{\pi}{3} $:

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

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

The coordinates are:

$$ \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) $$