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 values of $ \tan(\theta) $ at various angles and verify using the unit circle, we consider the following angles: $ \theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} $1. For $ \theta = \frac{\pi}{4} $:$...
Answer 1 To determine the coordinates of a point on the unit circle where the tangent line has a slope of $\frac{3}{4}$, we start with the equation of the unit circle:$x^2 + y^2 = 1$The slope of the tangent line at a point $(x, y)$ on the circle can...
Answer 1 To evaluate the integral of $ \cos^3(x)\sin(x) $ with respect to $ x $, we use a substitution method:Let $ u = \cos(x) $, then $ du = -\sin(x) dx $. Consequently:$ \int \cos^3(x)\sin(x) dx = \int u^3 (-du) = -\int u^3 du $Now integrate:$...
Answer 1 To find the angle in radians with an $x$-coordinate of $\frac{1}{2}$ in the first quadrant, we use the unit circle definition of cosine.For $\cos(\theta) = \frac{1}{2}$, the corresponding angle is:$ \theta = \frac{\pi}{3} $Answer 2 Given...
Answer 1 Given an angle $ \theta = \frac{7\pi}{6} $ radians, we need to determine its cosine and convert the angle to degrees.\nFirst, convert the angle to degrees:\n \n$ \theta = \frac{7\pi}{6} \cdot \frac{180^\circ}{\pi} = 210^\circ $\nThe angle $...
Answer 1 To find the value of $ \sec(\theta) $ for $ \theta $ in the unit circle, we need to recall the definition of secant. The secant function is the reciprocal of the cosine function: $ \sec(\theta) = \frac{1}{\cos(\theta)} $ Given that $ \theta...