Use reference triangles and unit circle to write the given trigonometric composition as an algebraic function: $$\sin \left( \cot ^ { - 1 } ( x ) \right)$$
A) $\frac { 1 } { \sqrt { 1 - x ^ { 2 } } }$
B) $\frac { 1 } { \sqrt { 1 + x ^ { 2 } } }$
C) $\frac { x } { \sqrt { 1 - x ^ { 2 } } }$
D) $\frac { x } { \sqrt { 1 - x ^ { 2 } } }$

Question

Use reference triangles and unit circle to write the given trigonometric composition as an algebraic function: $$\sin \left( \cot ^ { - 1 } ( x ) \right)$$
A) $\frac { 1  } { \sqrt { 1 - x ^ { 2 } } }$
B) $\frac {  1 } { \sqrt { 1 + x ^ { 2 } } }$
C) $\frac { x } { \sqrt { 1 - x ^ {  2 } } }$
D) $\frac { x } { \sqrt { 1 - x ^ { 2  } } }$

Quizplus · Accepted Answer

To solve $\sin(\cot^{-1}(x))$, consider a right triangle where the cotangent of an angle $\theta$ equals $x$, which means $\cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} = x$. This implies the adjacent side is $x$ and the opposite side is $1$. Using the Pythagorean theorem, the hypotenuse is $\sqrt{x^2 + 1}$. Therefore, $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{x^2 + 1}}$, which matches choice B.

Use Reference Triangles and Unit Circle to Write the Given sin⁡(cot⁡−1(x))\sin \left( \cot ^ { - 1 } ( x ) \right)sin(cot−1(x))

Use Reference Triangles and Unit Circle to Write the Given $\sin \left( \cot ^ { - 1 } ( x ) \right)$