Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

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

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

Step 5

Separate fractions.

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

Step 6

Convert from $\frac{\sin (t)}{\cos (t)}$ to $\tan (t)$ .

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

Step 7

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

$\frac{1}{\sec (t) - \cos (t)} \tan (t)$

Step 8

Combine $\frac{1}{\sec (t) - \cos (t)}$ and $\tan (t)$ .

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

Step 9

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 10

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 11

Substitute the actual values of $a = \frac{\tan (t)}{\sec (t) - \cos (t)}$ and $b = 0$ .

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

Step 12

Find $| z |$ .

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Step 12.1

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

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

Step 12.2

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

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

Step 12.3

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

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

Step 12.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 12.5

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

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

Step 12.6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 12.6.1

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

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

Step 12.6.2

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

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

Step 12.7

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

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

Step 12.8

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

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

Step 12.9

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

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

Step 12.10

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

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

Step 12.11

Separate fractions.

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

Step 12.12

Convert from $\frac{\sin (t)}{\cos (t)}$ to $\tan (t)$ .

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

Step 12.13

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

$| z | = \frac{1}{\sec (t) - \cos (t)} \cdot \tan (t)$

Step 12.14

Combine $\frac{1}{\sec (t) - \cos (t)}$ and $\tan (t)$ .

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

Step 13

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

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

Step 14

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

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