Math

Determine the coordinates on the unit circle where the tangent line passes through the point (1,2)

To find the points on the unit circle where the tangent line passes through the point $(1, 2)$, we can use the following steps:

1. The equation of the unit circle is given by $$x^2 + y^2 = 1$$.

2. The slope of the tangent line at any point $(a, b)$ on the circle is $$-a/b$$.

3. The equation of the tangent line at $(a, b)$ can be written as:

$$y – b = -\frac{a}{b}(x – a)$$

4. Substitute the point $(1, 2)$ into this equation to find $(a, b)$:

$$2 – b = -\frac{a}{b}(1 – a)$$

5. Simplify and solve for $(a, b)$.

After solving, we get:

$$a = \frac{\sqrt{2}}{3}$$ and $$b = \frac{\sqrt{2}}{3}$$.

Thus, the coordinates are: $$(\frac{\sqrt{2}}{3}, \frac{\sqrt{2}}{3})$$ and $$(\frac{-\sqrt{2}}{3}, \frac{-\sqrt{2}}{3})$$.

What is the cosine of an angle in the unit circle if the sine is negative?

In the unit circle, if the $\sin(\theta)$ is negative, it means that the angle $\theta$ is in the third or fourth quadrant.

In both of these quadrants, the sine value is negative.

Cosine values in these quadrants can be positive (fourth quadrant) or negative (third quadrant).

Therefore, the $\cos(\theta)$ can be expressed as:

$$\cos(\theta) = \pm\sqrt{1 – \sin^2(\theta)}$$

Determine the position of -π/2 on a unit circle

To find the position of $ -\frac{\pi}{2} $ on a unit circle, we start by understanding that the unit circle is a circle with radius 1 centered at the origin (0,0). The angle $ -\frac{\pi}{2} $ is measured in the clockwise direction from the positive x-axis.

In standard position, the angle $ -\frac{\pi}{2} $ corresponds to the point where the terminal side intersects the unit circle. This is the negative y-axis.

Thus, the coordinates of this point are:

$$ (0, -1) $$

Determine the values of theta where sin(theta) and cos(theta) are equal in the flipped unit circle

To determine the values of $ \theta $ where $ \sin(\theta) $ and $ \cos(\theta) $ are equal in the flipped unit circle, we start by setting up the equation:

$$ \sin(\theta) = \cos(\theta) $$

Dividing both sides by $ \cos(\theta) $, we get:

$$ \tan(\theta) = 1 $$

In the standard unit circle, $ \tan(\theta) = 1 $ when $ \theta = \frac{\pi}{4} + k\pi $, where $ k $ is an integer. However, since this is a flipped unit circle, we need to consider transformations:

$$ \theta = -\left(\frac{\pi}{4} + k\pi \right) $$

Hence, the values of $ \theta $ are given by:

$$ \theta = -\frac{\pi}{4} – k\pi $$

Determine which quadrant the angle pi/3 is in the unit circle

To determine the quadrant of the angle $ \frac{\pi}{3} $, we note that this angle is equivalent to 60 degrees.

In the unit circle, angles between 0 and 90 degrees are in the first quadrant.

Therefore, the angle $ \frac{\pi}{3} $ is in the first quadrant.

Determine the coordinates of points on the unit circle at specific angles

The unit circle is the circle of radius 1 centered at the origin (0, 0) in the coordinate plane. The coordinates of any point on the unit circle can be determined using trigonometric functions, specifically sine and cosine.

Given an angle $$\theta$$, the coordinates of the point on the unit circle are:

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

For example, for an angle $$\theta = 0$$, the coordinates are:

$$ (\cos(0), \sin(0)) = (1, 0) $$

For an angle $$\theta = \frac{\pi}{2}$$, the coordinates are:

$$ (\cos(\frac{\pi}{2}), \sin(\frac{\pi}{2})) = (0, 1) $$

Lastly, for an angle $$\theta = \pi$$, the coordinates are:

$$ (\cos(\pi), \sin(\pi)) = (-1, 0) $$

Find the sine and cosine of π/4

The unit circle helps us to memorize common angle values. For $ \frac{\pi}{4} $, the coordinates are the same for both sine and cosine.

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

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

Find the coordinates of the point on the unit circle where the angle is θ = π/4

The unit circle has a radius of 1. The coordinates of a point on the unit circle can be found using the formulas:

$$ x = \cos(\theta) $$

$$ y = \sin(\theta) $$

For $ \theta = \frac{\pi}{4} $:

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

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

So, the coordinates are:

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

What are the values of sin, cos, and tan for the angle π/4 on the unit circle?

To find the values of $ \sin $, $ \cos $, and $ \tan $ for the angle $ \frac{\pi}{4} $ on the unit circle, we use the unit circle properties:

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

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

$$ \tan \left( \frac{\pi}{4} \right) = \frac{ \sin \left( \frac{\pi}{4} \right) }{ \cos \left( \frac{\pi}{4} \right) } = 1 $$