Trigonometry Examples

$1 + \tan^{2} (x)$

Step 1

Rearrange terms.

$\tan^{2} (x) + 1$

Step 2

Apply pythagorean identity.

$\sec^{2} (x)$

Step 3

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 4

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 5

Substitute the actual values of $a = \sec^{2} (x)$ and $b = 0$ .

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

Step 6

Find $| z |$ .

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Step 6.1

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

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

Step 6.2

Multiply the exponents in ${(\sec^{2} (x))}^{2}$ .

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Step 6.2.1

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

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

Step 6.2.2

Multiply $2$ by $2$ .

$| z | = \sqrt{0 + \sec^{4} (x)}$

Step 6.3

Add $0$ and $\sec^{4} (x)$ .

$| z | = \sqrt{\sec^{4} (x)}$

Step 6.4

Rewrite $\sec^{4} (x)$ as ${(\sec^{2} (x))}^{2}$ .

$| z | = \sqrt{{(\sec^{2} (x))}^{2}}$

Step 6.5

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

$| z | = \sec^{2} (x)$

Step 7

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

$θ = \arctan (\frac{0}{\sec^{2} (x)})$

Step 8

Substitute the values of $θ = \arctan (\frac{0}{\sec^{2} (x)})$ and $| z | = \sec^{2} (x)$ .

$\sec^{2} (x) (\cos (\arctan (\frac{0}{\sec^{2} (x)})) + i \sin (\arctan (\frac{0}{\sec^{2} (x)})))$