Determine the value of theta for which the point (cos(θ), sin(θ)) on the unit circle forms a right-angled triangle with the origin and the point (1, 0)

by zhanhongyuan | Feb 6, 2025 | Unit Circle

Home Determine the value of $ heta $ for which the point $ (cos( heta), sin( heta)) $ on the unit circle forms a right-angled triangle with the origin and the point $ (1, 0) $ Answer 1 William King Given the points $ (cos(theta), sin(theta))$, the origin $(0, 0)$, and...

Calculate the area of a sector in a unit circle with a given central angle θ

by zhanhongyuan | Feb 6, 2025 | Unit Circle

Home Calculate the area of a sector in a unit circle with a given central angle $ heta $. Answer 1 Maria Rodriguez To calculate the area of a sector in a unit circle with a given central angle $ theta $ (in radians), use the following formula: $ A = frac{1}{2} cdot...

Find the values of tan(θ), sin(θ), and cos(θ) for θ = 45 degrees

by zhanhongyuan | Feb 6, 2025 | Unit Circle

Home Find the values of $ an( heta)$, $sin( heta)$, and $cos( heta)$ for $ heta = 45^circ$ Answer 1 Thomas Walker To find the values of $tan(theta)$, $sin(theta)$, and $cos(theta)$ for $theta = 45^circ$:First, we note that $theta = 45^circ$ is in the first quadrant of...

Prove the identity involving cos and sin on the unit circle

by zhanhongyuan | Feb 6, 2025 | Unit Circle

Home Prove the identity involving $cos( heta)$ and $sin( heta)$ on the unit circle Answer 1 Ava Martin To prove the identity involving $ cos(theta) $ and $ sin(theta) $ on the unit circle, we start with the Pythagorean identity:$ cos^2(theta) + sin^2(theta) = 1...

Find the coordinates of the point where the line y = 1 intersects the unit circle

by zhanhongyuan | Feb 6, 2025 | Unit Circle

Home Find the coordinates of the point where the line $ y = 1 $ intersects the unit circle Answer 1 Emily Hall To find the coordinates where the line $ y = 1 $ intersects the unit circle, we start by recalling the equation of the unit circle:$ x^2 + y^2 = 1...

Determine the equation of the tangent line to the unit circle at the point (1/2, sqrt(3)/2)

by zhanhongyuan | Feb 6, 2025 | Math

Home Determine the equation of the tangent line to the unit circle at the point $ left( frac{1}{2}, frac{sqrt{3}}{2} ight) $ Answer 1 Christopher Garcia The equation of the unit circle is given by: $ x^2 + y^2 = 1 $ To find the tangent line at the point $left(...

Find the exact values of sin, cos, and tan at 30 degrees on the unit circle

by zhanhongyuan | Feb 6, 2025 | Unit Circle

Home Find the exact values of $ sin $, $ cos $, and $ an $ at $ 30^{circ} $ on the unit circle Answer 1 Abigail Nelson First, we need to convert $ 30^{circ} $ to radians: $ 30^{circ} = frac{pi}{6} $ Using the unit circle, the coordinates for $ frac{pi}{6} $ are $...

Determine the value of tan(θ) when sin(θ) = 3/5 and θ is in the first quadrant

by zhanhongyuan | Feb 6, 2025 | Unit Circle

Home Determine the value of $ an(θ)$ when $sin(θ) = frac{3}{5}$ and $θ$ is in the first quadrant. Answer 1 Emily Hall Given that $sin(θ) = frac{3}{5}$ and $θ$ is in the first quadrant, we can find $cos(θ)$ using the Pythagorean identity:$sin^2(θ) + cos^2(θ) =...

Find the values of tan(θ) at various angles and verify using the unit circle

by zhanhongyuan | Feb 6, 2025 | Unit Circle

Home Find the values of $ an( heta) $ at various angles and verify using the unit circle Answer 1 Charlotte Davis 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},...

Determine the coordinates of a point on the unit circle where the tangent line has a slope of 3/4

by zhanhongyuan | Feb 6, 2025 | Unit Circle

Home Determine the coordinates of a point on the unit circle where the tangent line has a slope of $frac{3}{4}$ Answer 1 Joseph Robinson 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...