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Answer 1 Consider a unit circle with center at the origin (0,0). Let the endpoints of the chord be at coordinates (cos θ, sin θ) and (cos φ, sin φ). The formula for finding the distance between two points (x1, y1) and (x2, y2) is given by:$ d =...
Answer 1 Let's consider an angle $ \theta $ in the unit circle. The coordinates of a point on the unit circle are given by $(\cos \theta, \sin \theta)$. The tangent of the angle $ \theta $ is defined as:$\tan \theta = \frac{\sin \theta}{\cos...
Answer 1 Given a point on the unit circle, say $(\cos(\theta), \sin(\theta))$, we need to find the tangent line at this point.Step 1: The equation of the unit circle is $x^2 + y^2 = 1$.Step 2: To find the slope of the tangent, we differentiate...
Answer 1 To determine the tangent values for angles $\frac{\pi}{4}$, $\frac{2\pi}{3}$, and $\frac{5\pi}{6}$ on the unit circle, follow these steps:1. For the angle $\frac{\pi}{4}$: $\tan \left( \frac{\pi}{4} \right) = 1$2. For the angle...
Answer 1 Given the Pythagorean identity for the unit circle: $ x^2 + y^2 = 1 $ where $ x = \frac{3}{5}$, substitute this value into the identity: $ \left( \frac{3}{5} \right)^2 + y^2 = 1 $ $ \frac{9}{25} + y^2 = 1 $ Subtract $ \frac{9}{25}$ from both...
Answer 1 To find the angle whose cosine is $-\frac{2}{3}$, we need to look at the unit circle and identify the angles where the x-coordinate (cosine value) is $-\frac{2}{3}$. Since cosine is negative in the second and third quadrants, we look in...