Explore the unit circle and its relationship to angles, radians, trigonometric ratios, and coordinates in the coordinate plane.
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Answer 1 The general equation of a circle with center at $(h, k)$ and radius $r$ is: $(x - h)^2 + (y - k)^2 = r^2$ Here, $h = 3$, $k = 4$, and $r = 5$. Substitute these values into the equation: $(x - 3)^2 + (y - 4)^2 = 5^2$ Simplifying further: $(x...
Answer 1 To find the sine of the angle $\theta$ when $\theta = \frac{\pi}{6}$ radians:Step 1: Locate $\frac{\pi}{6}$ on the unit circle. The angle $\frac{\pi}{6}$ is 30 degrees.Step 2: Use the definition of sine on the unit circle, which is the...
Answer 1 Given a point on the unit circle corresponding to an angle of \( \frac{\pi}{6} \) (30°), determine the sine of the angle.The unit circle has a radius of 1. For an angle of \( \frac{\pi}{6} \), the coordinates are: $ \left( \cos...
Answer 1 To find the coordinates of the point on the unit circle at an angle of $\frac{\pi}{3}$ radians, we use the cosine and sine functions.The coordinates are given by: $ (\cos(\frac{\pi}{3}), \sin(\frac{\pi}{3})) $First, calculate the cosine: $...
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Answer 1 To find angles θ where $\sin(\theta) = \frac{1}{2}$ and $\cos(\theta) = -\frac{\sqrt{3}}{2}$, we start by identifying possible angles for each trigonometric condition separately:From $\sin(\theta) = \frac{1}{2}$, the possible angles are...