Precalculus Examples

$(\frac{π}{2}, π)$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{π}{2}, π)$ , draw the triangle between the three points $(0, 0)$ , $(\frac{π}{2}, 0)$ , and $(\frac{π}{2}, π)$ .

Opposite : $π$

Adjacent : $\frac{π}{2}$

Step 2

Find the hypotenuse using Pythagorean theorem $c = \sqrt{a^{2} + b^{2}}$ .

Tap for more steps...

Step 2.1

Apply the product rule to $\frac{π}{2}$ .

$\sqrt{\frac{π^{2}}{2^{2}} + {(π)}^{2}}$

Step 2.2

Raise $2$ to the power of $2$ .

$\sqrt{\frac{π^{2}}{4} + {(π)}^{2}}$

Step 2.3

To write $π^{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{\frac{π^{2}}{4} + π^{2} \cdot \frac{4}{4}}$

Step 2.4

Combine $π^{2}$ and $\frac{4}{4}$ .

$\sqrt{\frac{π^{2}}{4} + \frac{π^{2} \cdot 4}{4}}$

Step 2.5

Combine the numerators over the common denominator.

$\sqrt{\frac{π^{2} + π^{2} \cdot 4}{4}}$

Step 2.6

Rewrite $\frac{π^{2} + π^{2} \cdot 4}{4}$ in a factored form.

Tap for more steps...

Step 2.6.1

Move $4$ to the left of $π^{2}$ .

$\sqrt{\frac{π^{2} + 4 \cdot π^{2}}{4}}$

Step 2.6.2

Add $π^{2}$ and $4 π^{2}$ .

$\sqrt{\frac{5 π^{2}}{4}}$

Step 2.7

Rewrite $\sqrt{\frac{5 π^{2}}{4}}$ as $\frac{\sqrt{5 π^{2}}}{\sqrt{4}}$ .

$\frac{\sqrt{5 π^{2}}}{\sqrt{4}}$

Step 2.8

Simplify the numerator.

Tap for more steps...

Step 2.8.1

Reorder $5$ and $π^{2}$ .

$\frac{\sqrt{π^{2} \cdot 5}}{\sqrt{4}}$

Step 2.8.2

Pull terms out from under the radical.

$\frac{π \sqrt{5}}{\sqrt{4}}$

Step 2.9

Simplify the denominator.

Tap for more steps...

Step 2.9.1

Rewrite $4$ as $2^{2}$ .

$\frac{π \sqrt{5}}{\sqrt{2^{2}}}$

Step 2.9.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{π \sqrt{5}}{2}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{π}{\frac{π \sqrt{5}}{2}}$ .

$\frac{π}{\frac{π \sqrt{5}}{2}}$

Step 4

Simplify $\sin (θ)$ .

Tap for more steps...

Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\sin (θ) = π (\frac{2}{π \sqrt{5}})$

Step 4.2

Cancel the common factor of $π$ .

Tap for more steps...

Step 4.2.1

Factor $π$ out of $π \sqrt{5}$ .

$\sin (θ) = π (\frac{2}{π (\sqrt{5})})$

Step 4.2.2

Cancel the common factor.

$\sin (θ) = π (\frac{2}{π \sqrt{5}})$

Step 4.2.3

Rewrite the expression.

$\sin (θ) = \frac{2}{\sqrt{5}}$

Step 4.3

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sin (θ) = \frac{2}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 4.4

Combine and simplify the denominator.

Tap for more steps...

Step 4.4.1

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sin (θ) = \frac{2 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 4.4.2

Raise $\sqrt{5}$ to the power of $1$ .

$\sin (θ) = \frac{2 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 4.4.3

Raise $\sqrt{5}$ to the power of $1$ .

$\sin (θ) = \frac{2 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 4.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (θ) = \frac{2 \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 4.4.5

Add $1$ and $1$ .

$\sin (θ) = \frac{2 \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 4.4.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Tap for more steps...

Step 4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sin (θ) = \frac{2 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 4.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sin (θ) = \frac{2 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 4.4.6.3

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

$\sin (θ) = \frac{2 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 4.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.4.6.4.1

Cancel the common factor.

$\sin (θ) = \frac{2 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 4.4.6.4.2

Rewrite the expression.

$\sin (θ) = \frac{2 \sqrt{5}}{5}$

Step 4.4.6.5

Evaluate the exponent.

$\sin (θ) = \frac{2 \sqrt{5}}{5}$

Step 5

Approximate the result.

$\sin (θ) = \frac{2 \sqrt{5}}{5} \approx 0.89442719$