Math

Find the sine, cosine, and tangent values for the angle $\frac{\pi}{6}$ on the unit circle

To solve this, we need to find the sine, cosine, and tangent values for the angle $\frac{\pi}{6}$ on the unit circle.

The angle $\frac{\pi}{6}$ corresponds to 30 degrees.

Using the unit circle, we know that:

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

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

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

So, the values are:

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

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

$$ \tan \left( \frac{\pi}{6} \right) = \frac{\sqrt{3}}{3} $$

Find the value of sec(θ) when the terminal point of angle θ lies on the unit circle at coordinates (1/2, √3/2)

Given the coordinates $\left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right)$ on the unit circle, we know that the x-coordinate represents $\cos(\theta)$. Therefore:

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

The secant function is the reciprocal of the cosine function:

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

Substitute $ \cos(\theta)$ with $\frac{1}{2}$:

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

Therefore, the value of $ \sec(\theta)$ is 2.

Find the value of sec(θ) when the point on the unit circle corresponding to θ is (1/2, √3/2)

Given the point on the unit circle corresponding to $\\theta$ is $ (\\frac{1}{2}, \\frac{\\sqrt{3}}{2})$, we know

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

Therefore,

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

Methods to Quickly Memorize the Unit Circle

$$Method\ 1: \ Use\ Symmetry$$

$$Explanation: \ The\ unit\ circle\ is\ symmetric\ about\ the\ x-axis,\ y-axis,\ and\ the\ origin.\ By\ knowing\ the\ key\ points\ in\ the\ first\ quadrant,\ you\ can\ easily\ deduce\ the\ corresponding\ points\ in\ the\ other\ three\ quadrants.\ Specifically,\ remember\ coordinates\ of\ (\frac{\pi}{6},\ \frac{\sqrt{3}}{2},\ \frac{1}{2})\ and\ (\frac{\pi}{4},\ \frac{\sqrt{2}}{2},\ \frac{\sqrt{2}}{2})\ and\ (\frac{\pi}{3},\ \frac{1}{2},\ \frac{\sqrt{3}}{2}) \ for\ first\ quadrant.\ Rest\ will\ be\ just\ reflections. $$

Find the cosine and sine of pi/10 on the unit circle

To find the cosine and sine of $ \frac{\pi}{10} $ on the unit circle, we use the following steps.

Since $ \frac{\pi}{10} $ is an angle in radians, we can find its coordinates on the unit circle. The coordinates of an angle $ \theta $ on the unit circle are given by $ (\cos(\theta), \sin(\theta)) $.

Therefore, for $ \theta = \frac{\pi}{10} $:

$$ \cos(\frac{\pi}{10}) $$ $$ \cos(\frac{\pi}{10}) \approx 0.9511 $$

$$ \sin(\frac{\pi}{10}) $$ $$ \sin(\frac{\pi}{10}) \approx 0.3090 $$

Thus, the cosine and sine of $ \frac{\pi}{10} $ on the unit circle are approximately $ 0.9511 $ and $ 0.3090 $, respectively.

Find the sine, cosine, and tangent values for 45 degrees on the unit circle

To find the sine, cosine, and tangent values for $45^\circ$ on the unit circle, we use the following formulas:

$$\sin 45^\circ = \frac{1}{\sqrt{2}}$$

$$\cos 45^\circ = \frac{1}{\sqrt{2}}$$

$$\tan 45^\circ = 1$$

Therefore, the sine, cosine, and tangent values for $45^\circ$ are $\frac{1}{\sqrt{2}}$, $\frac{1}{\sqrt{2}}$, and $1$ respectively.

Learning the Unit Circle Easily

First, understand the basics of the unit circle. The unit circle is a circle with a radius of 1, centered at the origin (0, 0) of a coordinate plane.

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

Next, memorize key angles and their coordinates in both degrees and radians. For example:

$$0^{\circ} (0, 1)$$

$$90^{\circ} \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right)$$

$$180^{\circ} (0, -1)$$

$$270^{\circ} \left( -\frac{1}{2}, \frac{-\sqrt{3}}{2} \right)$$

Use symmetry to find coordinates of other angles.

Find the value of tan θ given that θ is an angle on the unit circle with a terminal side passing through the point (-1/2, -√3/2)

To find the value of $$\tan \theta $$, we use the fact that tan is defined as the ratio of the y-coordinate to the x-coordinate on the unit circle.

Given the point $$\left( -\frac{1}{2}, -\frac{\sqrt{3}}{2} \right)$$, we have:

$$\tan \theta = \frac{y}{x} = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}}$$

Simplify the expression:

$$\tan \theta = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \frac{\sqrt{3}}{1} = \sqrt{3}$$

Thus, the value of $$\tan \theta$$ is $$\sqrt{3}$$.

Calculate the cosine and sine of the angle pi/3 using the unit circle

Using the unit circle, we know that the angle $\frac{\pi}{3}$ corresponds to 60 degrees.

From the unit circle properties:

The coordinates at $\frac{\pi}{3}$ are $(\frac{1}{2}, \frac{\sqrt{3}}{2})$.

So, the cosine of $\frac{\pi}{3}$ is $\frac{1}{2}$ and the sine of $\frac{\pi}{3}$ is $\frac{\sqrt{3}}{2}$.

Answer: Cosine: $\frac{1}{2}$, Sine: $\frac{\sqrt{3}}{2}$.

What is the value of sin(30°) + cos(60°) + tan(45°) on the unit circle?

To solve for $\sin(30°) + \cos(60°) + \tan(45°)$, we need to find the individual values:

$$ \sin(30°) = \frac{1}{2} $$

$$ \cos(60°) = \frac{1}{2} $$

$$ \tan(45°) = 1 $$

Adding these values together:

$$ \sin(30°) + \cos(60°) + \tan(45°) = \frac{1}{2} + \frac{1}{2} + 1 = 2 $$

Therefore, the value is 2.