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 sine of the angle, we need to identify the y-coordinate of the given point $\left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$ on the unit circle.The y-coordinate is:$ -\frac{1}{2} $Therefore, the sine of the angle is:$ \sin(\theta)...
Answer 1 The angle $ \theta = 45^\circ $ is in the first quadrant.Answer 2 The angle $ heta = 135^circ $ is in the second quadrant.Answer 3 The angle $ heta = 225^circ $ is in the third quadrant.Start...
Answer 1 To find the secant line to the unit circle that is equidistant from the $x$-axis, we use the equation of the unit circle $ x^2 + y^2 = 1 $and the general equation of a line $ y = mx + b $Since the secant line is equidistant from the...
Answer 1 To find the exact values of sine and cosine for the angle $ \frac{\pi}{4} $, we use the unit circle.For $ \theta = \frac{\pi}{4} $, the coordinates on the unit circle are:$ ( \cos( \frac{\pi}{4} ), \sin( \frac{\pi}{4} )) $Since $...
Answer 1 To find the coordinates of specific angles on the unit circle, remember that the unit circle has a radius of 1.For the angle $\theta = \frac{\pi}{4}$, the coordinates are:$(\cos(\frac{\pi}{4}), \sin(\frac{\pi}{4})) = (\frac{\sqrt{2}}{2},...
Answer 1 The formula to find the area of a sector in a unit circle is:$ A = \frac{1}{2} \theta $where $ \theta $ is the central angle in radians.For example, if $ \theta = \frac{\pi}{4} $:$ A = \frac{1}{2} \times \frac{\pi}{4} = \frac{\pi}{8} $Answer...