Trigonometry Examples

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

Step 1

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

Opposite : $\frac{\sqrt{2}}{2}$

Adjacent : $\frac{\sqrt{2}}{2}$

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

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

Step 2.2.4.2

Rewrite the expression.

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

Step 2.2.5

Evaluate the exponent.

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

Step 2.3

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

$\sqrt{\frac{2}{4} + {(\frac{\sqrt{2}}{2})}^{2}}$

Step 2.4

Cancel the common factor of $2$ and $4$ .

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Step 2.4.1

Factor $2$ out of $2$ .

$\sqrt{\frac{2 (1)}{4} + {(\frac{\sqrt{2}}{2})}^{2}}$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $2$ out of $4$ .

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

Step 2.4.2.2

Cancel the common factor.

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

Step 2.4.2.3

Rewrite the expression.

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

Step 2.5

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

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

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

Step 2.6.5

Evaluate the exponent.

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

Step 2.7

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

$\sqrt{\frac{1}{2} + \frac{2}{4}}$

Step 2.8

Cancel the common factor of $2$ and $4$ .

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Step 2.8.1

Factor $2$ out of $2$ .

$\sqrt{\frac{1}{2} + \frac{2 (1)}{4}}$

Step 2.8.2

Cancel the common factors.

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Step 2.8.2.1

Factor $2$ out of $4$ .

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

Step 2.8.2.2

Cancel the common factor.

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

Step 2.8.2.3

Rewrite the expression.

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

Step 2.9

Simplify the expression.

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Step 2.9.1

Combine the numerators over the common denominator.

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

Step 2.9.2

Add $1$ and $1$ .

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

Step 2.9.3

Divide $2$ by $2$ .

$\sqrt{1}$

Step 2.9.4

Any root of $1$ is $1$ .

$1$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\frac{\sqrt{2}}{2}}{1}$ .

$\frac{\frac{\sqrt{2}}{2}}{1}$

Step 4

Divide $\frac{\sqrt{2}}{2}$ by $1$ .

$\sin (θ) = \frac{\sqrt{2}}{2}$

Step 5

Approximate the result.

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