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Find the exact values of sin, cos, and tan for the angle 225° using the unit circle
Answer 1 To find the exact values of $\sin$, $\cos$, and $\tan$ for the angle $225^{\circ}$ using the unit circle, we first note that $225^{\circ}$ is in the third quadrant. In the third quadrant, both sine and cosine values are negative, and tangent...
Finding the Tangent to a Unit Circle
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...
Given that the point on the unit circle corresponding to the angle θ is (-3/5, -4/5), determine the value of θ in radians, ensuring that θ lies within the interval [0, 2π) Describe your method and calculations in detail
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...
Given a point P on the unit circle at an angle θ (in radians) from the positive x-axis, find the coordinates of P if θ is transformed by the function f(θ) = 2θ + π/4 Also, identify the new x and y coordinates after the transformation
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...
Finding the Coordinates of a Point on the Unit Circle
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...
Find the circumference of a circle with a radius of 7 units
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...