Answer 1 First, recall that cosine represents the x-coordinate on the unit circle. The cosine of $ \theta $ equals $ -\frac{1}{2} $ at the angles $ \theta = \frac{2\pi}{3} $ and $ \theta = \frac{4\pi}{3} $ in radians. We can find these angles by...

Answer 1 We start by setting the equation:$\sin(\theta) = \cos(\theta)$We know from trigonometric identities that:$\sin(\theta) = \cos(\theta)$Dividing both sides by $\cos(\theta)$ (assuming $\cos(\theta) \neq 0$):$\frac{\sin(\theta)}{\cos(\theta)} =...

Answer 1 Let the point be $(x, y)$. Since it lies on the unit circle, we have $x^2 + y^2 = 1$. Consider the trigonometric equation $\cos(2\theta) + \sin(3\theta) = 0$. We know $\cos(2\theta) = 2\cos^2(\theta) - 1$ and $\sin(3\theta) = 3\sin(\theta) -...

Answer 1 First, let's find the coordinates of the angle $\frac{\pi}{4}$ on the unit circle. The unit circle has a radius of 1, and the coordinates at an angle $\theta$ are $(\cos(\theta), \sin(\theta))$.For $\theta =...

Answer 1 To find the points of intersection between the line $y = 2x + 3$ and the unit circle $x^2 + y^2 = 1$, we substitute $y = 2x + 3$ into the circle's equation: $ x^2 + (2x + 3)^2 = 1 $ Expanding and simplifying: $ x^2 + (4x^2 + 12x + 9) = 1 $ $...