Calculus Examples

$x = \sin (\frac{1}{2} t)$ , $y = \cos (\frac{1}{2} t)$

Step 1

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

$x = \sin (\frac{1}{2} t)$

Step 2

Rewrite the equation as $\sin (\frac{1}{2} t) = x$ .

$\sin (\frac{1}{2} t) = x$

Step 3

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

$\frac{1}{2} t = \arcsin (x)$

Step 4

Simplify the left side.

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Step 4.1

Combine $\frac{1}{2}$ and $t$ .

$\frac{t}{2} = \arcsin (x)$

Step 5

Multiply both sides of the equation by $2$ .

$2 \frac{t}{2} = 2 \arcsin (x)$

Step 6

Simplify the left side.

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Step 6.1

Cancel the common factor of $2$ .

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Step 6.1.1

Cancel the common factor.

$2 \frac{t}{2} = 2 \arcsin (x)$

Step 6.1.2

Rewrite the expression.

$t = 2 \arcsin (x)$

Step 7

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

$y = \cos (\frac{1}{2} \cdot (2 \arcsin (x)))$

Step 8

Remove parentheses.

$y = \cos (\frac{1}{2} \cdot (2 \arcsin (x)))$

Step 9

Simplify $\cos (\frac{1}{2} \cdot (2 \arcsin (x)))$ .

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Step 9.1

Reduce the expression by cancelling the common factors.

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Step 9.1.1

Cancel the common factor of $2$ .

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Step 9.1.1.1

Factor $2$ out of $2 \arcsin (x)$ .

$y = \cos (\frac{1}{2} \cdot (2 (\arcsin (x))))$

Step 9.1.1.2

Cancel the common factor.

$y = \cos (\frac{1}{2} \cdot (2 \arcsin (x)))$

Step 9.1.1.3

Rewrite the expression.

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

Step 9.1.2

Write the expression using exponents.

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Step 9.1.2.1

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, $\cos (\arcsin (x))$ is $\sqrt{1 - x^{2}}$ .

$y = \sqrt{1 - x^{2}}$

Step 9.1.2.2

Rewrite $1$ as $1^{2}$ .

$y = \sqrt{1^{2} - x^{2}}$

Step 9.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

$y = \sqrt{(1 + x) (1 - x)}$