by zhanhongyuan | Feb 2, 2025 | Unit Circle
Home Prove that the integral of $ frac{sin(x)}{x} $ from $ 0 $ to $ infty $ is $ frac{pi}{2} $ Answer 1 Lucas Brown To prove that the integral of $ frac{sin(x)}{x} $ from $ 0 $ to $ infty $ is $ frac{pi}{2} $, we will use the fact that:$ int_0^infty frac{sin(x)}{x} ,...
by zhanhongyuan | Feb 2, 2025 | Math
Home Find the equation of the circle that passes through the points $ A(1, 2) $, $ B(4, 6) $, and $ C(-3, -5) $ Answer 1 Lily Perez To find the equation of the circle passing through the points $ A(1, 2) $, $ B(4, 6) $, and $ C(-3, -5) $, we can use the determinant...
by zhanhongyuan | Feb 2, 2025 | Unit Circle
Home Solve for all $x$ given that $sin(x) + cos(2x) = 1$, where $x$ is an angle on the unit circle. Answer 1 John Anderson To solve for all $x$ given that $sin(x) + cos(2x) = 1$:First, use the double-angle identity for cosine: $cos(2x) = 2cos^2(x) – 1$Substitute...
by zhanhongyuan | Feb 2, 2025 | Unit Circle
Home Determine the exact value of a trigonometric expression involving radians on the unit circle Answer 1 Lucas Brown Consider the trigonometric expression $ cosleft(frac{7pi}{4}right) + sinleft(frac{7pi}{4}right) $. Determine its exact value using the unit...
by zhanhongyuan | Feb 2, 2025 | Unit Circle
Home Evaluate the integral of $ cos(x)sin(x) $ from $ 0 $ to $ frac{pi}{2} $ Answer 1 Emma Johnson To evaluate the integral of $ cos(x)sin(x) $ from $ 0 $ to $ frac{pi}{2} $, we can use the substitution method. Let:$ u = sin(x) $Then,$ du = cos(x) dx $The integral...
by zhanhongyuan | Feb 2, 2025 | Unit Circle
Home Explain how to derive the sine and cosine values of standard angles using the unit circle. Answer 1 Lucas Brown To derive the sine and cosine values of standard angles (0°, 30°, 45°, 60°, and 90°) using the unit circle, follow these steps:1. Draw the unit circle...
by zhanhongyuan | Feb 1, 2025 | Unit Circle
Home Evaluate the integral of the $ an( heta) $ function over the unit circle Answer 1 James Taylor To evaluate the integral of the function $ tan(theta) $ over the unit circle, we need to use the parametrization of the unit circle: $ x = cos(theta), quad y =...
by zhanhongyuan | Feb 1, 2025 | Unit Circle
Home Find the sine and cosine values for $ frac{5pi}{4} $ on the unit circle Answer 1 Joseph Robinson To find the sine and cosine values for $ frac{5pi}{4} $ on the unit circle, we need to locate the angle on the unit circle. The angle $ frac{5pi}{4} $ is in the third...
by zhanhongyuan | Feb 1, 2025 | Unit Circle
Home Calculate the cartesian coordinates of the intersection points of the unit circle and the line $ y = 2x + 1 $ Answer 1 Maria Rodriguez First, the equation of the unit circle is:$ x^2 + y^2 = 1 $Substitute $ y = 2x + 1 $ into $ x^2 + y^2 = 1 $:$ x^2 + (2x + 1)^2 =...
by zhanhongyuan | Feb 1, 2025 | Unit Circle
Home Find the exact value of $ sin(frac{pi}{4}) $ on the unit circle Answer 1 Joseph Robinson To find the exact value of $ sin(frac{pi}{4}) $ on the unit circle, recognize that $ frac{pi}{4} $ is 45 degrees. The sine of 45 degrees (or $ frac{pi}{4} $) is:$...