Trigonometry Examples

$\sin (\arccos (\frac{\sqrt{2}}{2}))$

Step 1

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

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

Step 2

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Step 3

Any root of $1$ is $1$ .

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

Step 4

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

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

Step 5

Combine and simplify the denominator.

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Step 5.1

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

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

Step 5.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.4

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

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

Step 5.5

Add $1$ and $1$ .

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

Step 5.6

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

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Step 5.6.1

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

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

Step 5.6.2

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

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

Step 5.6.3

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

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

Step 5.6.4

Cancel the common factor of $2$ .

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Step 5.6.4.1

Cancel the common factor.

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

Step 5.6.4.2

Rewrite the expression.

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

Step 5.6.5

Evaluate the exponent.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.70710678 \dots$