Math

Find the values of sin(θ), cos(θ), and tan(θ) for θ = 7π/6 using the unit circle

To find the values of $ \sin(\theta) $, $ \cos(\theta) $, and $ \tan(\theta) $ for $ \theta = \frac{7\pi}{6} $ using the unit circle, we start by locating the angle on the unit circle:

$ \theta = \frac{7\pi}{6} $ corresponds to an angle in the third quadrant, where both sine and cosine values are negative.

In the unit circle, for $ \theta = \frac{7\pi}{6} $:

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

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

To find the tangent, use: $$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $$

$$ \tan\left( \frac{7\pi}{6} \right) = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} $$

Find the value of cos(θ) when θ is on the Unit Circle at specific points

To find the value of $ \cos(\theta) $ on the Unit Circle at specific points, consider the following:

• When $ \theta = 0 $:

$$ \cos(0) = 1 $$

• When $ \theta = \frac{\pi}{2} $:

$$ \cos\left(\frac{\pi}{2}\right) = 0 $$

• When $ \theta = \pi $:

$$ \cos(\pi) = -1 $$

• When $ \theta = \frac{3\pi}{2} $:

$$ \cos\left(\frac{3\pi}{2}\right) = 0 $$

• When $ \theta = 2\pi $:

$$ \cos(2\pi) = 1 $$

Find the value of sin(θ) and cos(θ) at different points on the unit circle

To find the value of $ \sin(\theta) $ and $ \cos(\theta) $ at different points on the unit circle, consider the following angles:

1. $\theta = \frac{\pi}{6}$:

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

2. $\theta = \frac{\pi}{4}$:

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

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

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

Fill in the blank: The value of sin(____) at the unit circle when the angle is pi/2 is 1

Fill in the blank: The value of $\sin(____)$ at the unit circle when the angle is $\frac{\pi}{2}$ is 1.

Find the values of sine, cosine, and tangent of an angle theta when the angle is 225 degrees

To find the values of sine, cosine, and tangent of an angle $\theta$ when the angle is $225^\circ$, we use the unit circle:

The angle $225^\circ$ lies in the third quadrant, where sine and cosine are both negative:

$$\sin(225^\circ) = \sin(180^\circ + 45^\circ) = -\sin(45^\circ) = -\frac{\sqrt{2}}{2}$$

$$\cos(225^\circ) = \cos(180^\circ + 45^\circ) = -\cos(45^\circ) = -\frac{\sqrt{2}}{2}$$

$$\tan(225^\circ) = \frac{\sin(225^\circ)}{\cos(225^\circ)} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1$$

Calculate the sine and cosine values at 45 degrees using the unit circle

To calculate the sine and cosine values at $45^\circ$ using the unit circle, we recognize that a $45^\circ$ angle forms an isosceles right triangle in the unit circle.

The coordinates of the point where the angle intersects the unit circle are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.

Therefore, the values are:

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

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

Find the angle corresponding to the point (1/2, -√3/2) on the unit circle

To find the angle that corresponds to the point $ \left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right) $ on the unit circle, we look at the coordinates.

The x-coordinate is $ \frac{1}{2} $ and the y-coordinate is $ -\frac{\sqrt{3}}{2} $. These values correspond to an angle in the fourth quadrant.

The reference angle with these coordinates is $ \frac{\pi}{3} $ because:

$$ \cos \theta = \frac{1}{2} \text{ and } \sin \theta = -\frac{\sqrt{3}}{2} $$

Since the angle is in the fourth quadrant, the actual angle is:

$$ \theta = 2\pi – \frac{\pi}{3} = \frac{5\pi}{3} $$

Find the values of θ for which tan(θ) = 1 in the unit circle

To find the values of $ \theta $ for which $ \tan(\theta) = 1 $ on the unit circle, we need to identify the angles where the tangent function is equal to 1.

The tangent function is defined as the ratio of the sine and cosine functions:

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

For $ \tan(\theta) = 1 $, we have:

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

This implies:

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

On the unit circle, this equality occurs at:

$$ \theta = \frac{\pi}{4} + k\pi $$

where $ k $ is any integer. Therefore, the solutions are:

$$ \theta = \frac{\pi}{4}, \frac{5\pi}{4}, \frac{9\pi}{4}, \ldots $$

Find the exact values of the trigonometric functions for the angle 5π/3 in the unit circle

To find the exact values of the trigonometric functions for the angle $$ \frac{5\pi}{3} $$ in the unit circle, we first note that:

$$ \frac{5\pi}{3} = 2\pi – \frac{\pi}{3} $$

This means the angle is located in the fourth quadrant.

We can use the reference angle of $$ \frac{\pi}{3} $$:

$$ \cos \left( \frac{5\pi}{3} \right) = \cos \left( 2\pi – \frac{\pi}{3} \right) = \cos \left( \frac{\pi}{3} \right) = \frac{1}{2} $$

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

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

Determine the value of theta for which the point (cos(θ), sin(θ)) on the unit circle forms a right-angled triangle with the origin and the point (1, 0)

Given the points $ (\cos(\theta), \sin(\theta))$, the origin $(0, 0)$, and $(1, 0)$, we need to find $ \theta $ such that they form a right-angled triangle.

The distance between $(\cos(\theta), \sin(\theta))$ and $(1, 0)$ is:

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

Since $(\cos(\theta), \sin(\theta))$ lies on the unit circle, we use the Pythagorean identity:

$$ \cos^2(\theta) + \sin^2(\theta) = 1 $$

Thus the distance simplifies to:

$$ d = \sqrt{1 – 2\cos(\theta) + 1} = \sqrt{2 – 2\cos(\theta)} = \sqrt{2(1 – \cos(\theta))} $$

For the triangle to be right-angled, $\cos(\theta) = \frac{1}{2}$:

$$ \cos(\theta) = \frac{1}{2} \implies \theta = \frac{\pi}{3} \text{ or } \theta = -\frac{\pi}{3} $$