Trigonometry Examples

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

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 $a = \frac{2 + \tan^{2} (x)}{\sec^{2} (x)}$ and $b = - 1$ .

$| z | = \sqrt{{(- 1)}^{2} + {(\frac{2 + \tan^{2} (x)}{\sec^{2} (x)})}^{2}}$

Step 4

Find $| z |$ .

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Step 4.1

Raise $- 1$ to the power of $2$ .

$| z | = \sqrt{1 + {(\frac{2 + \tan^{2} (x)}{\sec^{2} (x)})}^{2}}$

Step 4.2

Apply the product rule to $\frac{2 + \tan^{2} (x)}{\sec^{2} (x)}$ .

$| z | = \sqrt{1 + \frac{{(2 + \tan^{2} (x))}^{2}}{{(\sec^{2} (x))}^{2}}}$

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

Multiply $2$ by $2$ .

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

Step 5

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

$θ = \arctan (\frac{- 1}{\frac{2 + \tan^{2} (x)}{\sec^{2} (x)}})$

Step 6

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

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