Trigonometry Examples

$(\sqrt{2}, \sqrt{2})$

Step 1

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

Opposite : $\sqrt{2}$

Adjacent : $\sqrt{2}$

Step 2

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

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

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

$\sqrt{{(2^{\frac{1}{2}})}^{2} + {(\sqrt{2})}^{2}}$

Step 2.1.2

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

$\sqrt{2^{\frac{1}{2} \cdot 2} + {(\sqrt{2})}^{2}}$

Step 2.1.3

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

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

Step 2.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.4.1

Cancel the common factor.

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

Step 2.1.4.2

Rewrite the expression.

$\sqrt{2^{1} + {(\sqrt{2})}^{2}}$

Step 2.1.5

Evaluate the exponent.

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

$\sqrt{2 + {(2^{\frac{1}{2}})}^{2}}$

Step 2.2.2

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

$\sqrt{2 + 2^{\frac{1}{2} \cdot 2}}$

Step 2.2.3

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

$\sqrt{2 + 2^{\frac{2}{2}}}$

Step 2.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.4.1

Cancel the common factor.

$\sqrt{2 + 2^{\frac{2}{2}}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{2 + 2^{1}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{2 + 2}$

Step 2.3

Simplify the expression.

Tap for more steps...

Step 2.3.1

Add $2$ and $2$ .

$\sqrt{4}$

Step 2.3.2

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

$\sqrt{2^{2}}$

Step 2.4

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

$2$

Step 3

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

$\frac{\sqrt{2}}{2}$

Step 4

Approximate the result.

$\sin (θ) = \frac{\sqrt{2}}{2} \approx 0.70710678$