Trigonometry Examples

$\frac{\sec (t) - \cos (t)}{\sin (t)}$

Step 1

Rewrite $\sec (t)$ in terms of sines and cosines.

$\frac{\frac{1}{\cos (t)} - \cos (t)}{\sin (t)}$

Step 2

Convert from $\frac{1}{\cos (t)}$ to $\sec (t)$ .

$\frac{\sec (t) - \cos (t)}{\sin (t)}$

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 $a = \frac{\sec (t) - \cos (t)}{\sin (t)}$ and $b = 0$ .

$| z | = \sqrt{0^{2} + {(\frac{\sec (t) - \cos (t)}{\sin (t)})}^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

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

$| z | = \sqrt{0 + {(\frac{\sec (t) - \cos (t)}{\sin (t)})}^{2}}$

Step 6.2

Rewrite $\sec (t)$ in terms of sines and cosines.

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

Step 6.3

Apply the product rule to $\frac{\frac{1}{\cos (t)} - \cos (t)}{\sin (t)}$ .

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

Step 6.4

Add $0$ and $\frac{{(\frac{1}{\cos (t)} - \cos (t))}^{2}}{\sin^{2} (t)}$ .

$| z | = \sqrt{\frac{{(\frac{1}{\cos (t)} - \cos (t))}^{2}}{\sin^{2} (t)}}$

Step 6.5

Rewrite $\frac{{(\frac{1}{\cos (t)} - \cos (t))}^{2}}{\sin^{2} (t)}$ as ${(\frac{\frac{1}{\cos (t)} - \cos (t)}{\sin (t)})}^{2}$ .

$| z | = \sqrt{{(\frac{\frac{1}{\cos (t)} - \cos (t)}{\sin (t)})}^{2}}$

Step 6.6

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

$| z | = \frac{\frac{1}{\cos (t)} - \cos (t)}{\sin (t)}$

Step 6.7

Convert from $\frac{1}{\cos (t)}$ to $\sec (t)$ .

$| z | = \frac{\sec (t) - \cos (t)}{\sin (t)}$

Step 7

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

$θ = \arctan (\frac{0}{\frac{\sec (t) - \cos (t)}{\sin (t)}})$

Step 8

Substitute the values of $θ = \arctan (\frac{0}{\frac{\sec (t) - \cos (t)}{\sin (t)}})$ and $| z | = \frac{\sec (t) - \cos (t)}{\sin (t)}$ .

$\frac{\sec (t) - \cos (t)}{\sin (t) (\cos (\arctan (\frac{0}{\frac{\sec (t) - \cos (t)}{\sin (t)}})) + i \sin (\arctan (\frac{0}{\frac{\sec (t) - \cos (t)}{\sin (t)}})))}$