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Answer 1 The unit circle helps us find the cosine of an angle. For an angle of $ \frac{3π}{4} $ radians:The reference angle is $ \x0crac{π}{4} $, and in the second quadrant, the cosine is negative.So, $ \cos(\frac{3π}{4}) = -\cos(\frac{π}{4}) $We...
Answer 1 To prove that the equation $ x^2 + y^2 = 1 $ is satisfied by the coordinates of any point on the unit circle for a given angle \theta , we start with the unit circle definition:\n On the unit circle, the coordinates of a point corresponding...
Answer 1 To evaluate $ \csc(\theta) $ when $ \sin(\theta) $ is a rational value, letAnswer 2 To evaluate $ csc( heta) $ given $ sin( heta) = frac{3}{5} $:Since $ csc( heta) = frac{1}{sin( heta)} $:$ csc( heta) = frac{5}{3} $Answer 3 Given $ sin(...
Answer 1 To find the coordinates of a point on the unit circle given an angle $ \theta $, we use the formulas for sine and cosine:\n $ x = \cos(\theta) $\n $ y = \sin(\theta) $\n For example, if $ \theta = \frac{\pi}{4} $:\n $ x =...
Answer 1 The angle \( \frac{3\pi}{4} \) is in the second quadrant of the unit circle. To find its coordinates, we start by noting that the reference angle for \( \frac{3\pi}{4} \) is \( \frac{\pi}{4} \). The coordinates for \( \frac{\pi}{4} \) are \(...
Answer 1 To calculate the length of the arc intercepted by a central angle $ \theta $ on a unit circle, you can use the formula: $ s = r \theta $ Since the radius $ r $ of the unit circle is 1, the formula simplifies to: $ s = \theta $ Thus, the...