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} $) is:$...
by zhanhongyuan | Feb 1, 2025 | Unit Circle
Home [et_pb_heading title=”Find the exact value of $ an( heta) $ on the unit circle for the angle $ heta $ where $ 0 < heta < frac{pi}{2} $ and $ heta $ is the solution to the equation $ 2sin( heta)cos( heta) = 1 $." _builder_version="4.27.3"...
by zhanhongyuan | Feb 1, 2025 | Unit Circle
Home What is the sine of $ frac{π}{6} $? Answer 1 Lucas Brown The sine of $ frac{π}{6} $ is found using the unit circle. At angle $ frac{π}{6} $, the y-coordinate of the corresponding point on the unit circle is: $ sin left( frac{π}{6} right) = frac{1}{2} $ Answer 2...
by zhanhongyuan | Feb 1, 2025 | Unit Circle
Home Evaluate the integral of $ frac{cos(2x)}{sqrt{1-sin^2(2x)}} $ with respect to $ x $ Answer 1 Henry Green To evaluate the integral $ int frac{cos(2x)}{sqrt{1-sin^2(2x)}} , dx $, we begin by recognizing that:$ sin^2(2x) + cos^2(2x) = 1 $Thus, the expression under...
by zhanhongyuan | Feb 1, 2025 | Unit Circle
Home Find the secant of an angle $ heta$ in a unit circle Answer 1 Benjamin Clark To find the secant of an angle $theta$ in a unit circle, we use the formula:$ sec(theta) = frac{1}{cos(theta)} $Suppose $theta$ is an angle in the first quadrant where cos(θ) = 0.6....