Precalculus Examples

$2 \cos^{2} (22.5 °) - 1$

Step 1

Apply the cosine double-angle identity.

$\cos (2 \cdot 22.5 °)$

Step 2

Multiply $2$ by $22.5 °$ .

$\cos (45)$

Step 3

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

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Step 3.1

Rewrite $45$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cos (\frac{90}{2})$

Step 3.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$\pm \sqrt{\frac{1 + \cos (90)}{2}}$

Step 3.3

Change the $\pm$ to $+$ because cosine is positive in the first quadrant.

$\sqrt{\frac{1 + \cos (90)}{2}}$

Step 3.4

The exact value of $\cos (90)$ is $0$ .

$\sqrt{\frac{1 + 0}{2}}$

Step 3.5

Simplify $\sqrt{\frac{1 + 0}{2}}$ .

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Step 3.5.1

Add $1$ and $0$ .

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

Step 3.5.2

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

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

Step 3.5.3

Any root of $1$ is $1$ .

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

Step 3.5.4

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

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

Step 3.5.5

Combine and simplify the denominator.

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Step 3.5.5.1

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

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

Step 3.5.5.2

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

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

Step 3.5.5.3

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

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

Step 3.5.5.4

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

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

Step 3.5.5.5

Add $1$ and $1$ .

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

Step 3.5.5.6

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

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Step 3.5.5.6.1

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 3.5.5.6.2

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

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

Step 3.5.5.6.3

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

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

Step 3.5.5.6.4

Cancel the common factor of $2$ .

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Step 3.5.5.6.4.1

Cancel the common factor.

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

Step 3.5.5.6.4.2

Rewrite the expression.

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

Step 3.5.5.6.5

Evaluate the exponent.

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

Step 4

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 5

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

Step 6

Substitute the actual values of $a = \frac{\sqrt{2}}{2}$ and $b = 0$ .

$| z | = \sqrt{0^{2} + {(\frac{\sqrt{2}}{2})}^{2}}$

Step 7

Find $| z |$ .

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Step 7.1

Apply basic rules of exponents.

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Step 7.1.1

Raising $0$ to any positive power yields $0$ .

$| z | = \sqrt{0 + {(\frac{\sqrt{2}}{2})}^{2}}$

Step 7.1.2

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

$| z | = \sqrt{0 + \frac{{\sqrt{2}}^{2}}{2^{2}}}$

Step 7.2

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

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Step 7.2.1

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

$| z | = \sqrt{0 + \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}}}$

Step 7.2.2

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

$| z | = \sqrt{0 + \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}}}$

Step 7.2.3

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

$| z | = \sqrt{0 + \frac{2^{\frac{2}{2}}}{2^{2}}}$

Step 7.2.4

Cancel the common factor of $2$ .

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Step 7.2.4.1

Cancel the common factor.

$| z | = \sqrt{0 + \frac{2^{\frac{2}{2}}}{2^{2}}}$

Step 7.2.4.2

Rewrite the expression.

$| z | = \sqrt{0 + \frac{2}{2^{2}}}$

Step 7.2.5

Evaluate the exponent.

$| z | = \sqrt{0 + \frac{2}{2^{2}}}$

Step 7.3

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

$| z | = \sqrt{0 + \frac{2}{4}}$

Step 7.4

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

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Step 7.4.1

Factor $2$ out of $2$ .

$| z | = \sqrt{0 + \frac{2 (1)}{4}}$

Step 7.4.2

Cancel the common factors.

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Step 7.4.2.1

Factor $2$ out of $4$ .

$| z | = \sqrt{0 + \frac{2 \cdot 1}{2 \cdot 2}}$

Step 7.4.2.2

Cancel the common factor.

$| z | = \sqrt{0 + \frac{2 \cdot 1}{2 \cdot 2}}$

Step 7.4.2.3

Rewrite the expression.

$| z | = \sqrt{0 + \frac{1}{2}}$

Step 7.5

Add $0$ and $\frac{1}{2}$ .

$| z | = \sqrt{\frac{1}{2}}$

Step 7.6

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

$| z | = \frac{\sqrt{1}}{\sqrt{2}}$

Step 7.7

Any root of $1$ is $1$ .

$| z | = \frac{1}{\sqrt{2}}$

Step 7.8

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

$| z | = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 7.9

Combine and simplify the denominator.

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Step 7.9.1

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

$| z | = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 7.9.2

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

$| z | = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 7.9.3

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

$| z | = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 7.9.4

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

$| z | = \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 7.9.5

Add $1$ and $1$ .

$| z | = \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 7.9.6

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

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Step 7.9.6.1

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

$| z | = \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 7.9.6.2

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

$| z | = \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 7.9.6.3

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

$| z | = \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 7.9.6.4

Cancel the common factor of $2$ .

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Step 7.9.6.4.1

Cancel the common factor.

$| z | = \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 7.9.6.4.2

Rewrite the expression.

$| z | = \frac{\sqrt{2}}{2}$

Step 7.9.6.5

Evaluate the exponent.

$| z | = \frac{\sqrt{2}}{2}$

Step 8

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

$θ = \arctan (\frac{0}{\frac{\sqrt{2}}{2}})$

Step 9

Since inverse tangent of $\frac{0}{\frac{\sqrt{2}}{2}}$ produces an angle in the first quadrant, the value of the angle is $0$ .

$θ = 0$

Step 10

Substitute the values of $θ = 0$ and $| z | = \frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2 (\cos (0) + i \sin (0))}$