Math

Find the values of sine, cosine, and tangent for the angle 5π/6 using the unit circle

To find the values of $ \sin \frac{5\pi}{6} $, $ \cos \frac{5\pi}{6} $, and $ \tan \frac{5\pi}{6} $ using the unit circle, we first locate the angle $ \frac{5\pi}{6} $ on the unit circle.

The angle $ \frac{5\pi}{6} $ is in the second quadrant, where sine is positive and cosine is negative. The reference angle is $ \pi – \frac{5\pi}{6} = \frac{\pi}{6} $.

From the unit circle, we know:

$$ \sin \frac{\pi}{6} = \frac{1}{2} $$ and $$ \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} $$

Since we are in the second quadrant:

$$ \sin \frac{5\pi}{6} = \frac{1}{2} $$

$$ \cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2} $$

$$ \tan \frac{5\pi}{6} = \frac{ \sin \frac{5\pi}{6} }{ \cos \frac{5\pi}{6} } = \frac{ \frac{1}{2} }{ -\frac{\sqrt{3}}{2} } = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} $$

Find the points where the line y = mx + b is tangent to the unit circle

To find the points where the line $ y = mx + b $ is tangent to the unit circle, we start with the equation of the unit circle:

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

Substitute $ y = mx + b $ into the unit circle equation:

$$ x^2 + (mx + b)^2 = 1 $$

Expand the equation:

$$ x^2 + m^2x^2 + 2mxb + b^2 = 1 $$

Combine like terms:

$$ (1 + m^2)x^2 + 2mxb + b^2 = 1 $$

This is a quadratic equation in $ x $:

$$ (1 + m^2)x^2 + 2mxb + (b^2 – 1) = 0 $$

For the line to be tangent to the circle, the discriminant must be zero:

$$ (2mb)^2 – 4(1 + m^2)(b^2 – 1) = 0 $$

Simplify the discriminant:

$$ 4m^2b^2 – 4(1 + m^2)(b^2 – 1) = 0 $$

Solving this will give the condition on $ b $.

Find the value of tan(θ) when θ is at the angle π/4 on the unit circle

To find the value of $ \tan(\theta) $ when $ \theta $ is at the angle $ \frac{\pi}{4} $ on the unit circle, we use the definition of tangent in terms of sine and cosine:

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

At $ \theta = \frac{\pi}{4} $, we know:

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

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

So:

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

Find the exact values of sin(θ) and cos(θ) for θ = π/4

To find the exact values of $ \sin(\theta) $ and $ \cos(\theta) $ for $ \theta = \frac{\pi}{4} $, we use the unit circle definition:

On the unit circle, the coordinates of the point corresponding to $ \theta = \frac{\pi}{4} $ are:

$$ ( \cos(\frac{\pi}{4}), \sin(\frac{\pi}{4}) ) $$

For $ \theta = \frac{\pi}{4} $, both $ \sin(\frac{\pi}{4}) $ and $ \cos(\frac{\pi}{4}) $ are:

$$ \frac{\sqrt{2}}{2} $$

Thus, we have:

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

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

Find the tangent of an angle $\theta$ on the unit circle in different quadrants

To find the tangent of an angle $\theta$ on the unit circle, note that $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. Consider the following angles:

1. For $\theta = \frac{\pi}{4}$ in the first quadrant:

$$\tan\left(\frac{\pi}{4}\right) = \frac{\sin\left(\frac{\pi}{4}\right)}{\cos\left(\frac{\pi}{4}\right)} = 1$$

2. For $\theta = \frac{3\pi}{4}$ in the second quadrant:

$$\tan\left(\frac{3\pi}{4}\right) = \frac{\sin\left(\frac{3\pi}{4}\right)}{\cos\left(\frac{3\pi}{4}\right)} = -1$$

3. For $\theta = \frac{5\pi}{4}$ in the third quadrant:

$$\tan\left(\frac{5\pi}{4}\right) = \frac{\sin\left(\frac{5\pi}{4}\right)}{\cos\left(\frac{5\pi}{4}\right)} = 1$$

4. For $\theta = \frac{7\pi}{4}$ in the fourth quadrant:

$$\tan\left(\frac{7\pi}{4}\right) = \frac{\sin\left(\frac{7\pi}{4}\right)}{\cos\left(\frac{7\pi}{4}\right)} = -1$$

Determine the tangent values for points on the unit circle at angles θ = π/4, θ = 2π/3, and θ = 5π/6

To find the tangent values for the given angles on the unit circle, we use the definition of the tangent function, which is $ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $.

For $ \theta = \frac{\pi}{4} $:

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

Therefore,

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

For $ \theta = \frac{2\pi}{3} $:

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

Therefore,

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

For $ \theta = \frac{5\pi}{6} $:

$$ \sin\left(\frac{5\pi}{6}\right) = \frac{1}{2} \text{ and } \cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2} $$

Therefore,

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

Calculate the coordinates of a point on the unit circle, given theta in radians

To calculate the coordinates of a point on the unit circle given an angle $\theta$ in radians, use the formulas:

$$ x = \cos(\theta) $$

and

$$ y = \sin(\theta) $$

For example, if $\theta = \frac{\pi}{4}$, then:

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

and

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

Therefore, the coordinates are:

$$ \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) $$

Find the coordinates where the function f(theta) = sin(2theta) intersects with the unit circle

To find the coordinates where $ f(\theta) = \sin(2\theta) $ intersects the unit circle, we start by setting $ \sin(2\theta) = y $.

The unit circle equation is $ x^2 + y^2 = 1 $.

Since $ y = \sin(2\theta) $, we have $ x^2 + \sin^2(2\theta) = 1 $.

Using the double angle identity, $ \sin(2\theta) = 2\sin(\theta)\cos(\theta) $, we rewrite the equation:

$$ x^2 + 4\sin^2(\theta)\cos^2(\theta) = 1 $$

Next, let $ u = \sin(\theta) $ and $ v = \cos(\theta) $ so the equation becomes:

$$ x^2 + 4uv = 1 $$

We need to satisfy both $ u^2 + v^2 = 1 $ and $ x^2 + 4uv = 1 $. Solving for $ x $ and substituting values:

After solving, we find the coordinates where $ f(\theta) $ intersects the unit circle are:

$$ (x_1, y_1) = (\sqrt{1 – \sin^2(2\theta)}, \sin(2\theta)) $$

$$ (x_2, y_2) = (-\sqrt{1 – \sin^2(2\theta)}, \sin(2\theta)) $$

Find the sine and cosine values at π/3

To find the sine and cosine values at $ \frac{\pi}{3} $:

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The unit circle values for $ \frac{\pi}{3} $ are:

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

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