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Answer 1 To determine the quadrant of the angle $\frac{5\pi}{3}$ on the unit circle: 1. Identify the reference angle: $\frac{5\pi}{3} - 2\pi = \frac{-\pi}{3}$, which is equal to $\frac{\pi}{3}$. 2. Determine the quadrant where $\frac{5\pi}{3}$ lies:...
Answer 1 Given the angle $\theta = \frac{5\pi}{3}$, we need to find the cosine value. The unit circle coordinates at an angle $\theta$ are given by $(\cos(\theta), \sin(\theta))$. For $\theta = \frac{5\pi}{3}$, the angle is in the fourth quadrant...
Answer 1 To find the points of intersection, we can substitute $y = 2x + 1$ into the equation of the unit circle, which is $x^2 + y^2 = 1$.$x^2 + (2x + 1)^2 = 1$Expanding the equation:$x^2 + (4x^2 + 4x + 1) = 1$Combining like terms:$5x^2 + 4x + 1 =...
Answer 1 $\text{Given } \cos(\theta) = 0.5$$\text{We know that } \cos(\theta) = 0.5 \text{ at } \theta = \frac{\pi}{3} \text{ and } \theta = -\frac{\pi}{3} \text{ (or equivalently } \theta = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3}...
Answer 1 To find the exact values of $\sin(\theta)$ and $\cos(\theta)$ using the unit circle, let's consider $\theta = \frac{5\pi}{6}$.First, we know that $\frac{5\pi}{6}$ is in the second quadrant.In the second quadrant, sine is positive and cosine...
Answer 1 To determine the equation for a unit circle flipped over the y-axis, we start with the standard unit circle equation: $x^2 + y^2 = 1$ When we flip the unit circle over the y-axis, we change the sign of the x-coordinate. Therefore, the new...