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 points where the line $ y = mx + b $ is tangent to the unit circle, we start with the equation of the unit circle:$ x^2 + y^2 = 1 $Substitute $ y = mx + b $ into the unit circle equation:$ x^2 + (mx + b)^2 = 1 $Expand the...
Answer 1 To find the value of $ \tan(\theta) $ when $ \theta $ is at the angle $ \frac{\pi}{4} $ on the unit circle, we use the definition of tangent in terms of sine and cosine:$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $At $ \theta =...
Answer 1 To find the exact values of $ \sin(\theta) $ and $ \cos(\theta) $ for $ \theta = \frac{\pi}{4} $, we use the unit circle definition:On the unit circle, the coordinates of the point corresponding to $ \theta = \frac{\pi}{4} $ are:$ (...
Answer 1 To find the tangent of an angle $\theta$ on the unit circle, note that $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. Consider the following angles: 1. For $\theta = \frac{\pi}{4}$ in the first quadrant: $\tan\left(\frac{\pi}{4}\right)...
Answer 1 To find the tangent values for the given angles on the unit circle, we use the definition of the tangent function, which is $ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $.For $ \theta = \frac{\pi}{4} $:$ \sin\left(\frac{\pi}{4}\right)...
Answer 1 To find the equation of the circle passing through three points, we use the general form of the equation of a circle: $ (x - h)^2 + (y - k)^2 = r^2 $We substitute each point into the equation to get three equations with variables $ h $, $ k...