Calculus Examples

$x = \sin (t)$ , $y = \csc (t)$

Step 1

Set up the parametric equation for $x (t)$ to solve the equation for $t$ .

$x = \sin (t)$

Step 2

Rewrite the equation as $\sin (t) = x$ .

$\sin (t) = x$

Step 3

Take the inverse sine of both sides of the equation to extract $t$ from inside the sine.

$t = \arcsin (x)$

Step 4

Replace $t$ in the equation for $y$ to get the equation in terms of $x$ .

$y = \csc (\arcsin (x))$

Step 5

Remove parentheses.

$y = \csc (\arcsin (x))$

Step 6

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - x^{2}}, x)$ , $(\sqrt{1^{2} - x^{2}}, 0)$ , and the origin. Then $\arcsin (x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - x^{2}}, x)$ . Therefore, $\csc (\arcsin (x))$ is $\frac{1}{x}$ .

$y = \frac{1}{x}$