Precalculus Examples

$\sin (w) = \frac{\sqrt{3}}{2}$ , $\sin (\frac{w}{2})$

Step 1

Use the definition of sine to find the known sides of the unit circle right triangle. The quadrant determines the sign on each of the values.

$\sin (w) = \frac{opposite}{hypotenuse}$

Step 2

Find the adjacent side of the unit circle triangle. Since the hypotenuse and opposite sides are known, use the Pythagorean theorem to find the remaining side.

$Adjacent = \sqrt{{hypotenuse}^{2} - {opposite}^{2}}$

Step 3

Replace the known values in the equation.

$Adjacent = \sqrt{{(2)}^{2} - {(\sqrt{3})}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

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

Adjacent $= \sqrt{4 - {(\sqrt{3})}^{2}}$

Step 4.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

Adjacent $= \sqrt{4 - {(3^{\frac{1}{2}})}^{2}}$

Step 4.2.2

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

Adjacent $= \sqrt{4 - 3^{\frac{1}{2} \cdot 2}}$

Step 4.2.3

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

Adjacent $= \sqrt{4 - 3^{\frac{2}{2}}}$

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

Adjacent $= \sqrt{4 - 3^{\frac{2}{2}}}$

Step 4.2.4.2

Rewrite the expression.

Adjacent $= \sqrt{4 - 3}$

Step 4.2.5

Evaluate the exponent.

Adjacent $= \sqrt{4 - 1 \cdot 3}$

Step 4.3

Multiply $- 1$ by $3$ .

Adjacent $= \sqrt{4 - 3}$

Step 4.4

Subtract $3$ from $4$ .

Adjacent $= \sqrt{1}$

Step 4.5

Any root of $1$ is $1$ .

Adjacent $= 1$

Step 5

Use the definition of sine to find the value of $\sin (w)$ .

$\sin (w) = \frac{opposite}{hypotenuse}$

Step 6

Substitute in the known values.

$\sin (w) = \frac{\sqrt{3}}{2}$

Step 7

Expand $\sin (\frac{w}{2})$ to $\pm \frac{\sqrt{2 (1 - \cos (w))}}{2}$ .

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Step 7.1

Apply the sine half-angle identity.

$\pm \sqrt{\frac{1 - \cos (w)}{2}}$

Step 7.2

Rewrite $\sqrt{\frac{1 - \cos (w)}{2}}$ as $\frac{\sqrt{1 - \cos (w)}}{\sqrt{2}}$ .

$\pm \frac{\sqrt{1 - \cos (w)}}{\sqrt{2}}$

Step 7.3

Multiply $\frac{\sqrt{1 - \cos (w)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\pm \frac{\sqrt{1 - \cos (w)}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 7.4

Combine and simplify the denominator.

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Step 7.4.1

Multiply $\frac{\sqrt{1 - \cos (w)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\pm \frac{\sqrt{1 - \cos (w)} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 7.4.2

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

$\pm \frac{\sqrt{1 - \cos (w)} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 7.4.3

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

$\pm \frac{\sqrt{1 - \cos (w)} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 7.4.4

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

$\pm \frac{\sqrt{1 - \cos (w)} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 7.4.5

Add $1$ and $1$ .

$\pm \frac{\sqrt{1 - \cos (w)} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 7.4.6

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

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Step 7.4.6.1

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

$\pm \frac{\sqrt{1 - \cos (w)} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 7.4.6.2

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

$\pm \frac{\sqrt{1 - \cos (w)} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 7.4.6.3

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

$\pm \frac{\sqrt{1 - \cos (w)} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 7.4.6.4

Cancel the common factor of $2$ .

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Step 7.4.6.4.1

Cancel the common factor.

$\pm \frac{\sqrt{1 - \cos (w)} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 7.4.6.4.2

Rewrite the expression.

$\pm \frac{\sqrt{1 - \cos (w)} \sqrt{2}}{2^{1}}$

Step 7.4.6.5

Evaluate the exponent.

$\pm \frac{\sqrt{1 - \cos (w)} \sqrt{2}}{2}$

Step 7.5

Combine using the product rule for radicals.

$\pm \frac{\sqrt{(1 - \cos (w)) \cdot 2}}{2}$

Step 7.6

Reorder factors in $\pm \frac{\sqrt{(1 - \cos (w)) \cdot 2}}{2}$ .

$\pm \frac{\sqrt{2 (1 - \cos (w))}}{2}$

Step 8

Use the definition of $c o s$ to find the value of $\cos (w)$ . In this case, $\cos (w) = \frac{1}{2}$ .

$\cos (w) = \frac{1}{2}$

Step 9

Substitute the values into $\pm \frac{\sqrt{2 (1 - \cos (w))}}{2}$ .

$\sin (\frac{w}{2}) = \pm \frac{\sqrt{2 (1 - \frac{1}{2})}}{2}$

Step 10

Simplify the numerator.

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Step 10.1

Write $1$ as a fraction with a common denominator.

$\sin (\frac{w}{2}) = \pm \frac{\sqrt{2 (\frac{2}{2} - \frac{1}{2})}}{2}$

Step 10.2

Combine the numerators over the common denominator.

$\sin (\frac{w}{2}) = \pm \frac{\sqrt{2 (\frac{2 - 1}{2})}}{2}$

Step 10.3

Subtract $1$ from $2$ .

$\sin (\frac{w}{2}) = \pm \frac{\sqrt{2 (\frac{1}{2})}}{2}$

Step 10.4

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

$\sin (\frac{w}{2}) = \pm \frac{\sqrt{\frac{2}{2}}}{2}$

Step 10.5

Divide $2$ by $2$ .

$\sin (\frac{w}{2}) = \pm \frac{\sqrt{1}}{2}$

Step 10.6

Any root of $1$ is $1$ .

$\sin (\frac{w}{2}) = \pm \frac{1}{2}$