Unit Circle

Find the angle θ in the unit circle where the sum of sin(θ) and cos(θ) equals 15

To find the angle $ \theta $ where the sum of $ \sin(\theta) $ and $ \cos(\theta) $ equals 1.5, we start with the equation:

$$ \sin(\theta) + \cos(\theta) = 1.5 $$

We can use the Pythagorean identity:

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

Let’s square both sides of the original equation:

$$ (\sin(\theta) + \cos(\theta))^2 = 1.5^2 $$

This gives us:

$$ \sin^2(\theta) + 2\sin(\theta)\cos(\theta) + \cos^2(\theta) = 2.25 $$

Using the Pythagorean identity:

$$ 1 + 2\sin(\theta)\cos(\theta) = 2.25 $$

Therefore:

$$ 2\sin(\theta)\cos(\theta) = 1.25 $$

Which simplifies to:

$$ \sin(2\theta) = 1.25 $$

However, we know that the range of $ \sin(2\theta) $ is between -1 and 1, so no such $ \theta $ exists.

Find the sine and cosine values for an angle of pi/4 on the unit circle

To find the sine and cosine values for an angle of $ \frac{\pi}{4} $ on the unit circle, we can use the known values of the unit circle.

For an angle of $ \frac{\pi}{4} $:

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

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

Find the exact values of sin and cos for all solutions in the third quadrant for the equation 2*sin(x) + 3*cos(x) = 1

To find the exact values of $\sin(x)$ and $\cos(x)$ for all solutions in the third quadrant for the equation $2\sin(x) + 3\cos(x) = 1$, consider the trigonometric identity:

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

In the third quadrant, both $ \sin(x) $ and $ \cos(x) $ are negative. Let

Find the values of cos(θ) and sin(θ) at specific angles on the unit circle

Let

Find the value of tan(7π/6) and explain using the unit circle

To find the value of $ \tan(\frac{7\pi}{6}) $ using the unit circle:

1. Locate the angle $\frac{7\pi}{6}$ on the unit circle. This angle is in the third quadrant.

2. The reference angle for $\frac{7\pi}{6}$ is $\frac{\pi}{6}$.

3. In the third quadrant, both sine and cosine are negative. Knowing the coordinates for $\frac{\pi}{6}$ are $(\frac{\sqrt{3}}{2}, \frac{1}{2})$:

The coordinates for $\frac{7\pi}{6}$ are $(-\frac{\sqrt{3}}{2}, -\frac{1}{2})$.

4. Finally, calculate the tangent value:

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

Find the values of sin, cos, and tan for 45 degrees using the unit circle

To find the values of $ \sin $, $ \cos $, and $ \tan $ for $ 45^{\circ} $ using the unit circle, we start with:

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

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

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• $$ \tan(45^{\circ}) = 1 $$

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Determine the cosine of an angle corresponding to the point (1/2, sqrt(3)/2) on the unit circle

To determine the cosine of the angle corresponding to the point $ \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) $ on the unit circle, we must recognize the coordinates $(x, y)$ represent $(\cos(\theta), \sin(\theta))$.

In this case, the point is:

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

Thus, the cosine of the angle is:

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

Determine the sine and cosine values of an angle 𝜃 in radians on the unit circle given that 𝜃 = 5𝜋/4

Given $\theta = \frac{5\pi}{4}$, we determine the sine and cosine values by examining the unit circle.

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

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

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

Find the exact values of the inverse trigonometric functions on the unit circle

Consider the point $$P(-\frac{1}{2}, -\frac{\sqrt{3}}{2})$$ on the unit circle. Determine the exact values for the following inverse trigonometric functions:

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

2. $$\arccos(-\frac{1}{2})$$

3. $$\arctan(\frac{\sqrt{3}}{3})$$

Answer:

1. $$\arcsin(-\frac{\sqrt{3}}{2}) = -\frac{\pi}{3}$$

2. $$\arccos(-\frac{1}{2}) = \frac{2\pi}{3}$$

3. $$\arctan(\frac{\sqrt{3}}{3}) = \frac{\pi}{6}$$

Find the value of tan(θ) at θ = 3π/4 on the unit circle

To find the value of $ \tan(θ) $ at $ θ = \frac{3π}{4} $, we first identify the coordinates on the unit circle:

At $ θ = \frac{3π}{4} $, the coordinates are $ (-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) $.

So, $ \tan(θ) $ is given by:

$$ \tan(θ) = \frac{y}{x} = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = -1 $$

Therefore, $ \tan(θ) = -1 $.