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 circumference of a circle, we use the formula: $ C = 2\pi r $ Given that the radius \( r = 4 \) units, we substitute this value into the formula: $ C = 2 \pi \times 4 $ $ C = 8 \pi $ Therefore, the circumference is \( 8\pi \)...
Answer 1 On the unit circle, the coordinates for $\theta = \frac{\pi}{4}$ are $\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$. Therefore, $\tan(\frac{\pi}{4})$ is calculated as: $\tan(\frac{\pi}{4}) =...
Answer 1 Initial coordinates of point $P$ are $(\frac{\sqrt{3}}{2}, \frac{1}{2})$. When flipped over the $y$-axis, the x-coordinate becomes its negative value while the y-coordinate remains the same. Therefore, the new coordinates of point $P$...
Answer 1 The unit circle is a circle with a radius of 1 centered at the origin. To find the coordinates of a point at $45^{\circ}$, we use the trigonometric functions sine and cosine.For $\theta = 45^{\circ}$:$\cos(45^{\circ}) =...
Answer 1 To solve this problem, we start with the unit circle equation:$x^2 + y^2 = 1$Given that $\cos(\theta) = \frac{-3}{5}$, we know the x-coordinate is $\frac{-3}{5}$. Let's find the y-coordinate (sine value).Substituting $\cos(\theta)$ in the...
Answer 1 To find the Cartesian coordinates of a point on the unit circle where the angle is $135^{\circ}$, we use the unit circle equation:$x = \cos(135^{\circ})$$y = \sin(135^{\circ})$First, we calculate the cosine and sine of...