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 determine the angles $\theta$ where $\tan(\theta) = \sqrt{3}$, we first recognize that $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. From the unit circle, we know:$ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} $Thus, $\theta =...
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Answer 1 To determine the quadrant of a point $(x, y)$ on the unit circle, we need to consider the signs of $x$ and $y$. The unit circle is centered at the origin (0,0) and has a radius of 1.For example, let’s find which quadrant the point...
Answer 1 Given the unit circle equation: $\cos^2(θ) + \sin^2(θ) = 1$ Substitute $\cos(θ) = -\frac{1}{2}$: $\left(-\frac{1}{2}\right)^2 + \sin^2(θ) = 1$ Simplify: $\frac{1}{4} + \sin^2(θ) = 1$ Subtract $\frac{1}{4}$ from both sides: $\sin^2(θ) = 1 -...
Answer 1 Given $\theta = 45^\circ$, we need to find the values of $\sin(45^\circ)$, $\cos(45^\circ)$, and $\tan(45^\circ)$.From the unit circle, we know that $\sin(45^\circ) = \frac{\sqrt{2}}{2}$ and $\cos(45^\circ) = \frac{\sqrt{2}}{2}$.Using the...
Answer 1 Suppose we need to find the sine and cosine values for the angle $ \frac{\pi}{4} $ on the unit circle.Step 1: Recognize the angle $ \frac{\pi}{4} $ on the unit circle.Step 2: Recall that $ \frac{\pi}{4} $ corresponds to 45 degrees.Step 3:...