Convert to Trigonometric Form 5/(tan(x)+sec(x))

$\frac{5}{\tan (x) + \sec (x)}$

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

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

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

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

Substitute the actual values of $a = \frac{5}{\tan (x) + \sec (x)}$ and $b = 0$ .

$| z | = \sqrt{0^{2} + {(\frac{5}{\tan (x) + \sec (x)})}^{2}}$

Find $| z |$ .

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

$| z | = \sqrt{0 + {(\frac{5}{\tan (x) + \sec (x)})}^{2}}$

Apply the product rule to $\frac{5}{\tan (x) + \sec (x)}$ .

$| z | = \sqrt{0 + \frac{5^{2}}{{(\tan (x) + \sec (x))}^{2}}}$

Raise $5$ to the power of $2$ .

$| z | = \sqrt{0 + \frac{25}{{(\tan (x) + \sec (x))}^{2}}}$

Add $0$ and $\frac{25}{{(\tan (x) + \sec (x))}^{2}}$ .

$| z | = \sqrt{\frac{25}{{(\tan (x) + \sec (x))}^{2}}}$

Rewrite $25$ as $5^{2}$ .

$| z | = \sqrt{\frac{5^{2}}{{(\tan (x) + \sec (x))}^{2}}}$

Rewrite $\frac{5^{2}}{{(\tan (x) + \sec (x))}^{2}}$ as ${(\frac{5}{\tan (x) + \sec (x)})}^{2}$ .

$| z | = \sqrt{{(\frac{5}{\tan (x) + \sec (x)})}^{2}}$

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

$| z | = \frac{5}{\tan (x) + \sec (x)}$

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

$θ = \arctan (\frac{0}{\frac{5}{\tan (x) + \sec (x)}})$

Substitute the values of $θ = \arctan (\frac{0}{\frac{5}{\tan (x) + \sec (x)}})$ and $| z | = \frac{5}{\tan (x) + \sec (x)}$ .

$\frac{5}{(\tan (x) + \sec (x)) (\cos (\arctan (\frac{0}{\frac{5}{\tan (x) + \sec (x)}})) + i \sin (\arctan (\frac{0}{\frac{5}{\tan (x) + \sec (x)}})))}$

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