Math

Find the angle in radians corresponding to a point on the unit circle given coordinates (x, y)

To find the angle θ corresponding to the point $(\frac{1}{2}, \frac{\sqrt{3}}{2})$ on the unit circle, we start with the basic trigonometric relationships:

$$ \cos \theta = x $$

$$ \sin \theta = y $$

Given $ x = \frac{1}{2} $ and $ y = \frac{\sqrt{3}}{2} $, we can use the inverse trigonometric functions:

$$ \theta = \cos^{-1}\left(\frac{1}{2}\right) $$

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

We know that:

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

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

Therefore, the angle corresponding to the given point is:

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

Find the cosine value of the angle 120 degrees on the unit circle

$$120\ degrees\ is\ in\ the\ second\ quadrant.\ The\ reference\ angle\ is\ 180 – 120 = 60\ degrees.$$

$$In\ the\ second\ quadrant,\ \cos(\theta)\ is\ negative.$$

$$\cos(60\ degrees) = \frac{1}{2},\ so\ \cos(120\ degrees) = -\frac{1}{2}.$$

Find the values of sine, cosine, and tangent for a given angle on the unit circle

To find the values of sine, cosine, and tangent for the angle 𝜃 = $\frac{5\pi}{6}$:

1. Locate $\frac{5\pi}{6}$ on the unit circle. This angle corresponds to 150°.

2. Find the coordinates of the point on the unit circle at this angle. For $\frac{5\pi}{6}$, the coordinates are $(-\frac{\sqrt{3}}{2}, \frac{1}{2})$.

3. The cosine of the angle is the x-coordinate and the sine of the angle is the y-coordinate.

4. Tangent is given by $\tan \theta = \frac{\sin \theta}{\cos \theta}$.

Therefore:

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

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

$$\tan \left( \frac{5\pi}{6} \right) = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$$

Given that \( \theta \) is an angle in the unit circle such that its terminal side passes through the point (a,b) If the line passing through (a,b) and the origin makes an angle \( \alpha \) with the x-axis, find the values of \( \sin(\alpha) \), \( \cos

Given that \( \theta \) is an angle in the unit circle such that its terminal side passes through the point \((a,b)\):

The coordinates \((a, b)\) on the unit circle imply that \(a = \cos(\theta)\) and \(b = \sin(\theta)\).

Since the line passing through \((a, b)\) and the origin makes an angle \( \alpha \) with the x-axis:

$$ \sin(\alpha) = \frac{b}{\sqrt{a^2 + b^2}} $$

$$ \cos(\alpha) = \frac{a}{\sqrt{a^2 + b^2}} $$

$$ \tan(\alpha) = \frac{b}{a} $$

Given that \( \theta \) is in the second quadrant:

$$ \theta = \pi – \alpha $$

Find the sine and cosine of a 45-degree angle

To find the sine and cosine of a 45-degree angle, we can use the unit circle. A 45-degree angle corresponds to $\frac{\pi}{4}$ radians.

On the unit circle, the coordinates for $\frac{\pi}{4}$ are given by $\left( \cos \frac{\pi}{4}, \sin \frac{\pi}{4} \right)$.

We know from trigonometric identities that:

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

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

Therefore, the sine and cosine of a 45-degree angle are both $\frac{\sqrt{2}}{2}$.

What are the coordinates of the point on the unit circle where the angle is π/3?

To find the coordinates of the point on the unit circle at an angle of $\frac{\pi}{3}$ radians, we use the trigonometric functions cosine and sine.

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

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

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

Therefore, the coordinates are:

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

Find the value of tan(π/4) using the unit circle

To find the value of $ tan(\frac{\pi}{4}) $ using the unit circle, we need to consider the coordinates of the point on the unit circle at the angle $ \frac{\pi}{4} $. The coordinates of this point are $ (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) $.

The tangent function is defined as the ratio of the y-coordinate to the x-coordinate:

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

So,

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

Therefore, the value of $ tan(\frac{\pi}{4}) $ is 1.

Find the trigonometric values for an angle in the unit circle

Given an angle \( \theta = \frac{5\pi}{4} \), find the values of \( \sin(\theta) \), \( \cos(\theta) \), and \( \tan(\theta) \).

First, determine the reference angle in the unit circle. \( \theta = \frac{5\pi}{4} \) is in the third quadrant. The reference angle is \( \pi + \frac{\pi}{4} = \frac{5\pi}{4} \).

For the angle \( \frac{5\pi}{4} \):

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

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

\( \tan(\frac{5\pi}{4}) = 1 \)

Therefore, the values are:

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

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

\( \tan(\frac{5\pi}{4}) = 1 \)

Determining the Position of -π/2 on the Unit Circle

To locate the position of $-\pi/2$ on the unit circle, we need to understand the unit circle itself. The circle has a radius of 1 and is centered at the origin (0,0).

1. Start from the positive x-axis and move counterclockwise.

2. A negative angle indicates a clockwise direction.

So, $-\pi/2$ means we move $\pi/2$ radians clockwise from the positive x-axis.

At $-\pi/2$ radians, the coordinates on the unit circle are given by:

$$ (\cos(-\pi/2), \sin(-\pi/2)) $$

Since $\cos(-\pi/2) = 0$ and $\sin(-\pi/2) = -1$, the position is:

$$ (0, -1) $$

Determine the Value of sec(θ) Given the Coordinates on the Unit Circle

Given a point on the unit circle with coordinates (0.6, 0.8), determine the value of $\sec(\theta)$.

Step 1: Recall the definition of the point on the unit circle: $(\cos(\theta), \sin(\theta))$.

Thus, $\cos(\theta) = 0.6$.

Step 2: Recall the definition of secant in terms of cosine: $\sec(\theta) = \frac{1}{\cos(\theta)}$.

Step 3: Substitute $\cos(\theta)$ into the secant definition: $$\sec(\theta) = \frac{1}{0.6} = \frac{5}{3}$$.

Therefore, $\sec(\theta) = \frac{5}{3}$.