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Answer 1 The angle $ \frac{2\pi}{3} $ radians is in the second quadrant.In the second quadrant, the cosine of an angle is negative.For $ \frac{2\pi}{3} $ radians, the reference angle is $ \frac{\pi}{3} $ radians.Cosine of $ \frac{\pi}{3} $ radians is...
Answer 1 To prove that the integral of $ \exp(i \theta) $ over a complete unit circle is zero, we evaluate the contour integral: $ \int_0^{2\pi} \exp(i \theta) d\theta $ Recall that $ \exp(i \theta) = \cos(\theta) + i \sin(\theta) $. So the integral...
Answer 1 To find the sine and cosine values for an angle of $ \frac{π}{3} $ on the unit circle, we need to recall the special angles on the unit circle. For $ \frac{π}{3} $, the coordinates are (cos($ \frac{π}{3} $), sin($ \frac{π}{3} $)). Using the...
Answer 1 A unit circle is defined as a circle with a radius of $1$ unit.The diameter of a circle is twice the radius.Therefore, for a unit circle:$ \text{Diameter} = 2 \times \text{Radius} = 2 \times 1 = 2 $Thus, the diameter of a unit circle is $2$...
Answer 1 To find the tangent values at specific points on the unit circle, we use the definition of the tangent function, which is $ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $.1. At $ \theta = 0 $:$ \tan(0) = \frac{\sin(0)}{\cos(0)} =...
Answer 1 To find the coordinates of the point on the unit circle where the angle is $ \frac{\pi}{4} $ radians, we use the unit circle definition:The coordinates at this angle are:$ \left( \cos \frac{\pi}{4} , \sin \frac{\pi}{4} \right) $Using known...