Explore the unit circle and its relationship to angles, radians, trigonometric ratios, and coordinates in the coordinate plane.
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Answer 1 The equation for a unit circle centered at the origin in the Cartesian plane is: $ x^2 + y^2 = 1 $ This equation represents all points $(x, y)$ that are exactly one unit away from the origin.Answer 2 For a unit circle centered at the origin,...
Answer 1 To find the value of $ \tan(\frac{\pi}{4}) $ using the unit circle:On the unit circle, the coordinates for $ \frac{\pi}{4} $ are $ (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) $.Therefore:$ \tan(\frac{\pi}{4}) =...
Answer 1 To find the tangent line equations for every point on the unit circle, we start with the unit circle equation: $ x^2 + y^2 = 1 $ Differentiate implicitly with respect to $x$ to find the slope: $ 2x + 2y \x0crac{dy}{dx} = 0 $ Solve for $...
Answer 1 To find the value of $ \sec(\theta) $ for $ \theta = \frac{\pi}{4} $ on the unit circle, we use the definition of secant, which is the reciprocal of cosine:$ \sec(\theta) = \frac{1}{\cos(\theta)} $For $ \theta = \frac{\pi}{4} $, we have:$...
Answer 1 To find the measure of $\angle ABC$ given that the arc $\overset\frown{AC}$ is 120 degrees, we use the Inscribed Angle Theorem. The Inscribed Angle Theorem states that the measure of an inscribed angle is half the measure of its intercepted...
Answer 1 The unit circle is defined as the set of all points $(x, y)$ such that:\n$ x^2 + y^2 = 1 $\nIn the unit circle, the sine value corresponds to the y-coordinate and the cosine value corresponds to the x-coordinate. We need to find a point...