Precalculus Examples

$\cos (45 °)$

Step 1

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

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

Step 2

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 3

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 4

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

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

Step 5

Find $| z |$ .

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Step 5.1

Apply basic rules of exponents.

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.2

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

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Step 5.2.1

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 5.2.2

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 5.2.3

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

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

Step 5.2.4

Cancel the common factor of $2$ .

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Step 5.2.4.1

Cancel the common factor.

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

Step 5.2.4.2

Rewrite the expression.

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

Step 5.2.5

Evaluate the exponent.

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

Step 5.3

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

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

Step 5.4

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

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Step 5.4.1

Factor $2$ out of $2$ .

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

Step 5.4.2

Cancel the common factors.

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Step 5.4.2.1

Factor $2$ out of $4$ .

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

Step 5.4.2.2

Cancel the common factor.

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

Step 5.4.2.3

Rewrite the expression.

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

Any root of $1$ is $1$ .

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

Step 5.8

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

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

Step 5.9

Combine and simplify the denominator.

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Step 5.9.1

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

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

Step 5.9.2

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

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

Step 5.9.3

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

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

Step 5.9.4

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

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

Step 5.9.5

Add $1$ and $1$ .

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

Step 5.9.6

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

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Step 5.9.6.1

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 5.9.6.2

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

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

Step 5.9.6.3

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

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

Step 5.9.6.4

Cancel the common factor of $2$ .

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Step 5.9.6.4.1

Cancel the common factor.

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

Step 5.9.6.4.2

Rewrite the expression.

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

Step 5.9.6.5

Evaluate the exponent.

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

Step 6

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 7

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 8

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

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