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Answer 1 The problem is to find the equation of the tangent to a unit circle at a given point.Given a unit circle with the equation:$x^2 + y^2 = 1$and a point \((a, b)\) on the circle. Since \((a, b)\) is on the circle, we have:$a^2 + b^2 = 1$To find...
Answer 1 Given the point on the unit circle $\left( -\frac{3}{5}, -\frac{4}{5} \right)$, we need to determine the angle $\theta$ in radians.First, note that the x and y coordinates tell us which quadrant the angle is in. Both coordinates are...
Answer 1 Given the initial angle $ \theta $, the coordinates of the point $ P $ are: $ ( \cos \theta, \sin \theta ) $ With the transformation $ f(\theta) = 2\theta + \frac{\pi}{4} $, let the new angle be $ \theta' = 2\theta + \frac{\pi}{4} $. The new...
Answer 1 Given an angle of $\theta = \frac{\pi}{3}$ radians, find the coordinates of the corresponding point on the unit circle.First, recall the unit circle definition: for any angle $\theta$, the coordinates of the point on the unit circle are...
Answer 1 First, we use the formula for the circumference of a circle:$C = 2 \pi r$Substituting the radius (r) given in the problem:$C = 2 \pi \times 7$Simplify the expression:$C = 14\pi$The circumference of the circle is:$14\pi \text{ units}$Answer 2...
Answer 1 Consider the unit circle centered at the origin with the equation: $x^2 + y^2 = 1.$ To find the equation of the tangent line to the circle at a given point $P(a, b)$ on the circle, we follow these steps: 1. Verify that $P(a, b)$ lies on the...