Trigonometry Examples

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

Step 1

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 2

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 3

Substitute the actual values of $a = 1$ and $b = \sin^{2} (x)$ .

$| z | = \sqrt{{(\sin^{2} (x))}^{2} + 1^{2}}$

Step 4

Find $| z |$ .

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Step 4.1

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

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Step 4.1.1

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

$| z | = \sqrt{{\sin (x)}^{2 \cdot 2} + 1^{2}}$

Step 4.1.2

Multiply $2$ by $2$ .

$| z | = \sqrt{\sin^{4} (x) + 1^{2}}$

Step 4.2

One to any power is one.

$| z | = \sqrt{\sin^{4} (x) + 1}$

Step 5

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

$θ = \arctan (\frac{\sin^{2} (x)}{1})$

Step 6

Substitute the values of $θ = \arctan (\frac{\sin^{2} (x)}{1})$ and $| z | = \sqrt{\sin^{4} (x) + 1}$ .

$\sqrt{\sin^{4} (x) + 1} (\cos (\arctan (\frac{\sin^{2} (x)}{1})) + i \sin (\arctan (\frac{\sin^{2} (x)}{1})))$