Trigonometry Examples

$\cos (\frac{π}{3})$

Step 1

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\frac{1}{2}$

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 $a = \frac{1}{2}$ and $b = 0$ .

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

Step 5

Find $| z |$ .

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Step 5.1

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

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

Step 5.2

Apply the product rule to $\frac{1}{2}$ .

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

Step 5.3

One to any power is one.

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

Step 5.4

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

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

Step 5.5

Add $0$ and $\frac{1}{4}$ .

$| z | = \sqrt{\frac{1}{4}}$

Step 5.6

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$| z | = \frac{\sqrt{1}}{\sqrt{4}}$

Step 5.7

Any root of $1$ is $1$ .

$| z | = \frac{1}{\sqrt{4}}$

Step 5.8

Simplify the denominator.

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Step 5.8.1

Rewrite $4$ as $2^{2}$ .

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

Step 5.8.2

Pull terms out from under the radical, assuming positive real numbers.

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

Step 6

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

$θ = \arctan (\frac{0}{\frac{1}{2}})$

Step 7

Since inverse tangent of $\frac{0}{\frac{1}{2}}$ 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 $| z | = \frac{1}{2}$ .

$\frac{1}{2 (\cos (0) + i \sin (0))}$