Precalculus Examples

$x (t) = \sin (t) - 1$ , $y (t) = \csc (t)$

Step 1

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

$x = \sin (t) - 1$

Step 2

Rewrite the equation as $\sin (t) - 1 = x$ .

$\sin (t) - 1 = x$

Step 3

Add $1$ to both sides of the equation.

$\sin (t) = x + 1$

Step 4

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

$t = \arcsin (x + 1)$

Step 5

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

$y = \csc (\arcsin (x + 1))$

Step 6

Remove parentheses.

$y = \csc (\arcsin (x + 1))$

Step 7

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

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