Homework

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,

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

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 $$\sqrt{3}$$.

Locate -π/2 on a Unit Circle

To locate $-\pi/2$ on the unit circle, we can follow these steps:

1. Start at the positive x-axis (0 radians).

2. Move clockwise because the angle is negative.

3. Since $-\pi/2$ radians equals -90 degrees, move 90 degrees clockwise from the positive x-axis.

4. This will place you on the negative y-axis.

Therefore, the coordinates for $-\pi/2$ on the unit circle are (0, -1).

Suppose you have a unit circle centered at the origin in the coordinate plane You flip the unit circle over the y-axis Determine the coordinates of a point (x, y) on the original unit circle after the transformation, given that x^2 + y^2 = 1

$$\text{Given the equation of the original unit circle}$$

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

$$\text{When the unit circle is flipped over the y-axis, each point } (x, y) \text{ is transformed to } (-x, y).$$

$$\text{So, the new coordinates after transformation are } (-x, y).$$

$$\text{For instance, if you have a point } (x, y) = (\frac{1}{2}, \frac{\sqrt{3}}{2}) \text{ on the original unit circle, the transformed coordinates are:}$$

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

Find the exact values of sine and cosine for an angle of 5π/4 radians on the unit circle

To solve for sine and cosine of the angle $\frac{5\pi}{4}$, we first determine its location on the unit circle.

The angle $\frac{5\pi}{4}$ radians is in the third quadrant, where both sine and cosine values are negative.

The reference angle for $\frac{5\pi}{4}$ radians is $\pi/4$ radians, whose sine and cosine values are $\frac{\sqrt{2}}{2}$ and $\frac{\sqrt{2}}{2}$ respectively.

Thus, for $\frac{5\pi}{4}$:

$$\sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$$

$$\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$$

Find the value of sin(30 degrees) on the unit circle

To find the value of $\sin(30^\circ)$ on the unit circle, we first need to recognize that $30^\circ$ is a special angle. On the unit circle, the angle $30^\circ$ corresponds to the coordinates $(\frac{\sqrt{3}}{2}, \frac{1}{2})$. The sine function gives the y-coordinate of this point.

Therefore,

$$\sin(30^\circ) = \frac{1}{2}.$$

Techniques to Remember the Unit Circle for High School Students

One way to remember the unit circle is by focusing on the key angles and their coordinates. Let’s start with the four quadrants: $$0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$$ radians or $$0^\circ, 90^\circ, 180^\circ, 270^\circ, 360^\circ$$. The coordinates for these angles are as follows:

– $$0^\circ (1,0)$$

– $$90^\circ (0,1)$$

– $$180^\circ (-1,0)$$

– $$270^\circ (0,-1)$$

– $$360^\circ (1,0)$$

Find all angles θ in radians such that tan(θ) = 3 and θ is in the interval [0, 2π]

To solve the problem, we need to find all angles $\theta$ such that $\tan(\theta) = 3$ within the interval $[0, 2\pi]$.

Step 1: Recognize that $\tan(\theta)$ is positive in the first and third quadrants.

Step 2: The reference angle $\alpha$ for $\tan(\alpha) = 3$ is found using $\alpha = \arctan(3)$.

Step 3: Calculate $\alpha$:
$\alpha = \arctan(3) \approx 1.249$ radians.

Step 4: Identify the angles in the first and third quadrants:
$\theta_1 = \alpha = \arctan(3) \approx 1.249$ radians
$\theta_2 = \pi + \alpha = \pi + \arctan(3) \approx 4.391$ radians.

Therefore, the solutions are $\theta \approx 1.249$ radians and $\theta \approx 4.391$ radians.