Explore the unit circle and its relationship to angles, radians, trigonometric ratios, and coordinates in the coordinate plane.
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Answer 1 To find the values of $\tan(\theta)$, $\sin(\theta)$, and $\cos(\theta)$ for $\theta = 45^\circ$:First, we note that $\theta = 45^\circ$ is in the first quadrant of the unit circle.The coordinates of the point on the unit circle at $\theta =...
Answer 1 To prove the identity involving $ \cos(\theta) $ and $ \sin(\theta) $ on the unit circle, we start with the Pythagorean identity:$ \cos^2(\theta) + \sin^2(\theta) = 1 $Consider the parameterization of the unit circle with $ \theta $ as the...
Answer 1 To find the coordinates where the line $ y = 1 $ intersects the unit circle, we start by recalling the equation of the unit circle:$ x^2 + y^2 = 1 $Substituting $ y = 1 $ into the unit circle equation, we get:$ x^2 + 1^2 = 1 $Simplifying,$...
Answer 1 The equation of the unit circle is given by: $ x^2 + y^2 = 1 $ To find the tangent line at the point $\left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right)$, we need to determine the slope. Differentiating implicitly: $ 2x \frac{dx}{dt} + 2y...
Answer 1 First, we need to convert $ 30^{\circ} $ to radians: $ 30^{\circ} = \frac{\pi}{6} $ Using the unit circle, the coordinates for $ \frac{\pi}{6} $ are $ \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right) $ From this, we can find: $ \sin...
Answer 1 Given that $\sin(θ) = \frac{3}{5}$ and $θ$ is in the first quadrant, we can find $\cos(θ)$ using the Pythagorean identity:$\sin^2(θ) + \cos^2(θ) = 1$Plugging in the given value:$\left(\frac{3}{5}\right)^2 + \cos^2(θ) = 1$$\frac{9}{25} +...