Trigonometry Examples

$\sin (\arccos (\sqrt{\frac{3}{2}}))$

Step 1

Draw a triangle in the plane with vertices $(\sqrt{\frac{3}{2}}, \sqrt{1^{2} - {\sqrt{\frac{3}{2}}}^{2}})$ , $(\sqrt{\frac{3}{2}}, 0)$ , and the origin. Then $\arccos (\sqrt{\frac{3}{2}})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{\frac{3}{2}}, \sqrt{1^{2} - {\sqrt{\frac{3}{2}}}^{2}})$ . Therefore, $\sin (\arccos (\sqrt{\frac{3}{2}}))$ is $\sqrt{\frac{- 1}{2}}$ .

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

Step 2

Move the negative in front of the fraction.

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

Step 3

Rewrite $- \frac{1}{2}$ as $i^{2} \frac{1^{2}}{2}$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

One to any power is one.

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

Step 6

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

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

Step 7

Any root of $1$ is $1$ .

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

Step 8

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

$i (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 9

Combine and simplify the denominator.

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

Add $1$ and $1$ .

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

Step 9.6

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

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Step 9.6.1

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

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

Step 9.6.2

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

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

Step 9.6.3

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

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

Step 9.6.4

Cancel the common factor of $2$ .

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Step 9.6.4.1

Cancel the common factor.

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

Step 9.6.4.2

Rewrite the expression.

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

Step 9.6.5

Evaluate the exponent.

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

Step 10

Combine $i$ and $\frac{\sqrt{2}}{2}$ .

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