Math

Understanding the representation of sine on the unit circle

To understand what sine represents on the unit circle, let’s begin with the definition of the unit circle. The unit circle is a circle with a radius of 1 centered at the origin of the coordinate system.

Consider a point $P(x, y)$ on the unit circle that forms an angle $\theta$ with the positive x-axis. The coordinates of point $P$ can be expressed in terms of trigonometric functions as:

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

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

Therefore, the sine of the angle $\theta$ is the y-coordinate of the corresponding point on the unit circle.

To elaborate with a specific angle, let’s consider $\theta = \frac{\pi}{4}$. The coordinates of the point on the unit circle at this angle are:

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

Thus, for $\theta = \frac{\pi}{4}$, the sine value is:

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

This demonstrates that sine represents the y-coordinate of a point on the unit circle corresponding to a given angle.

Given a point on the unit circle at an angle theta in radians, determine the exact coordinates and verify their correctness for theta = 7π/6

We start with the unit circle formula:

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

Given $\theta = \frac{7\pi}{6}$, we need to find the cosine and sine of this angle:

$$\cos\left( \frac{7\pi}{6} \right) = \cos\left( \pi + \frac{\pi}{6} \right) = -\cos\left( \frac{\pi}{6} \right) = -\frac{\sqrt{3}}{2}$$

$$\sin\left( \frac{7\pi}{6} \right) = \sin\left( \pi + \frac{\pi}{6} \right) = -\sin\left( \frac{\pi}{6} \right) = -\frac{1}{2}$$

Thus, the coordinates are:

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

Verification:

$$\left( -\frac{\sqrt{3}}{2} \right)^2 + \left( -\frac{1}{2} \right)^2 = \frac{3}{4} + \frac{1}{4} = 1$$

The coordinates are correct.

Given a point P on the unit circle with coordinates (x, y), find the value of cotangent at angle θ where θ is the angle formed by the positive x-axis and the line segment OP

Given a point $P(x, y)$ on the unit circle:

$$x = \cos(\theta), \quad y = \sin(\theta)$$

The cotangent of angle $\theta$ is:

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

Since $x = \cos(\theta)$ and $y = \sin(\theta)$, we can write:

$$\cot(\theta) = \frac{x}{y}$$

Therefore, the value of $\cot(\theta)$ is:

$$\cot(\theta) = \frac{x}{y}$$

Finding the Coordinates of a Point on the Unit Circle Given an Angle

Given an angle \( \theta \) in radians, find the coordinates of the corresponding point on the unit circle.

Step 1: Recall the unit circle definition. The unit circle is a circle with a radius of 1 centered at the origin \((0, 0)\) in the coordinate plane.

Step 2: Use the cosine and sine functions to determine the coordinates: \(x = \cos(\theta)\), \(y = \sin(\theta)\).

Step 3: Compute the coordinates for a given angle \( \theta = \frac{5\pi}{6} \).

$$x = \cos \left( \frac{5\pi}{6} \right) = -\frac{\sqrt{3}}{2}$$

$$y = \sin \left( \frac{5\pi}{6} \right) = \frac{1}{2}$$

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

Find the Intersection Points on the Unit Circle

Consider the unit circle defined by the equation $x^2 + y^2 = 1$ and the line with the equation $y = mx + c$.

To find the intersection points, substitute $y = mx + c$ into the unit circle’s equation:

$$x^2 + (mx + c)^2 = 1$$

Expand and simplify:

$$x^2 + m^2x^2 + 2mxc + c^2 = 1$$

Combine like terms:

$$ (1 + m^2)x^2 + 2mxc + c^2 – 1 = 0 $$

This is a quadratic equation in $x$:

$$ Ax^2 + Bx + C = 0 $$

where $A = 1 + m^2$, $B = 2mc$, and $C = c^2 – 1$.

Solve for $x$ using the quadratic formula:

$$ x = \frac{-B \pm \sqrt{B^2 – 4AC}}{2A} $$

Substitute the values of $A$, $B$, and $C$:

$$ x = \frac{-2mc \pm \sqrt{(2mc)^2 – 4(1 + m^2)(c^2 – 1)}}{2(1 + m^2)} $$

Simplify further to find $x$, then use $y = mx + c$ to find the $y$ values of the intersection points.

Find the coordinates of the point on the unit circle corresponding to an angle of \(\theta = \frac{\pi}{4}\)

For an angle \(\theta = \frac{\pi}{4}\) on the unit circle, we use the trigonometric functions sine and cosine to find the coordinates. The coordinates are given by \((\cos \theta, \sin \theta)\).

$$ \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} $$

$$ \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} $$

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

Calculate the values of tan(π/4), tan(π/6), and tan(π/3) using the unit circle

Let’s calculate the values of $\tan(\pi/4)$, $\tan(\pi/6)$, and $\tan(\pi/3)$ using the unit circle:

1. $\tan(\pi/4)$:

On the unit circle, the angle $\pi/4$ (45 degrees) corresponds to the point $(\sqrt{2}/2, \sqrt{2}/2)$. The tangent function is defined as $\tan(\theta) = \frac{y}{x}$.

Therefore,

$$\tan(\pi/4) = \frac{\sqrt{2}/2}{\sqrt{2}/2} = 1$$

2. $\tan(\pi/6)$:

On the unit circle, the angle $\pi/6$ (30 degrees) corresponds to the point $(\sqrt{3}/2, 1/2)$. Therefore,

$$\tan(\pi/6) = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$$

3. $\tan(\pi/3)$:

On the unit circle, the angle $\pi/3$ (60 degrees) corresponds to the point $(1/2, \sqrt{3}/2)$. Therefore,

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

Given a point P on the unit circle at an angle of \( \theta = \frac{3\pi}{4} \), find the coordinates of point P Then, determine the value of \( \cos(2\theta) \) and \( \sin(2\theta) \)

When $ \theta = \frac{3\pi}{4} $, the coordinates of point P on the unit circle are given by $ (\cos(\theta), \sin(\theta)) $.

First, we need to calculate these values:

$$ \cos \left( \frac{3\pi}{4} \right) = -\frac{\sqrt{2}}{2} $$

$$ \sin \left( \frac{3\pi}{4} \right) = \frac{\sqrt{2}}{2} $$

So, the coordinates of point P are $ \left( -\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) $.

Next, we determine $ \cos(2\theta) $ and $ \sin(2\theta) $ using the double-angle formulas:

$$ \cos(2\theta) = \cos(2 \cdot \frac{3\pi}{4}) = \cos \left( \frac{6\pi}{4} \right) = \cos \left( \frac{3\pi}{2} \right) = 0 $$

$$ \sin(2\theta) = \sin(2 \cdot \frac{3\pi}{4}) = \sin \left( \frac{6\pi}{4} \right) = \sin \left( \frac{3\pi}{2} \right) = -1 $$

Find the coordinates of the point on the unit circle corresponding to an angle of \( \frac{5\pi}{4} \) radians

To find the coordinates of the point on the unit circle corresponding to the angle $ \frac{5\pi}{4} $ radians, we need to use the unit circle properties.

The angle $ \frac{5\pi}{4} $ radians is in the third quadrant.

The reference angle for $ \frac{5\pi}{4} $ is $ \pi/4 $ radians.

In the third quadrant, both sine and cosine values are negative.

From the unit circle, we know:

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

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

Therefore, the coordinates for $ \frac{5\pi}{4} $ are:

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