Trigonometry Examples

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

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 $a = \frac{1 - \sin (x)}{\cos (x)}$ and $b = 0$ .

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

Step 4

Find $| z |$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Multiply by $1$ .

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

Step 4.4

Separate fractions.

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

Step 4.5

Convert from $\frac{1}{\cos^{2} (x)}$ to $\sec^{2} (x)$ .

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

Step 4.6

Simplify the expression.

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Step 4.6.1

Divide ${(1 - \sin (x))}^{2}$ by $1$ .

$| z | = \sqrt{0 + {(1 - \sin (x))}^{2} \sec^{2} (x)}$

Step 4.6.2

Rewrite ${(1 - \sin (x))}^{2}$ as $(1 - \sin (x)) (1 - \sin (x))$ .

$| z | = \sqrt{0 + (1 - \sin (x)) (1 - \sin (x)) \sec^{2} (x)}$

Step 4.7

Expand $(1 - \sin (x)) (1 - \sin (x))$ using the FOIL Method.

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Step 4.7.1

Apply the distributive property.

$| z | = \sqrt{0 + (1 (1 - \sin (x)) - \sin (x) (1 - \sin (x))) \sec^{2} (x)}$

Step 4.7.2

Apply the distributive property.

$| z | = \sqrt{0 + (1 \cdot 1 + 1 (- \sin (x)) - \sin (x) (1 - \sin (x))) \sec^{2} (x)}$

Step 4.7.3

Apply the distributive property.

$| z | = \sqrt{0 + (1 \cdot 1 + 1 (- \sin (x)) - \sin (x) \cdot 1 - \sin (x) (- \sin (x))) \sec^{2} (x)}$

Step 4.8

Simplify and combine like terms.

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Step 4.8.1

Simplify each term.

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Step 4.8.1.1

Multiply $1$ by $1$ .

$| z | = \sqrt{0 + (1 + 1 (- \sin (x)) - \sin (x) \cdot 1 - \sin (x) (- \sin (x))) \sec^{2} (x)}$

Step 4.8.1.2

Multiply $- \sin (x)$ by $1$ .

$| z | = \sqrt{0 + (1 - \sin (x) - \sin (x) \cdot 1 - \sin (x) (- \sin (x))) \sec^{2} (x)}$

Step 4.8.1.3

Multiply $- 1$ by $1$ .

$| z | = \sqrt{0 + (1 - \sin (x) - \sin (x) - \sin (x) (- \sin (x))) \sec^{2} (x)}$

Step 4.8.1.4

Multiply $- \sin (x) (- \sin (x))$ .

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Step 4.8.1.4.1

Multiply $- 1$ by $- 1$ .

$| z | = \sqrt{0 + (1 - \sin (x) - \sin (x) + 1 \sin (x) \sin (x)) \sec^{2} (x)}$

Step 4.8.1.4.2

Multiply $\sin (x)$ by $1$ .

$| z | = \sqrt{0 + (1 - \sin (x) - \sin (x) + \sin (x) \sin (x)) \sec^{2} (x)}$

Step 4.8.1.4.3

Raise $\sin (x)$ to the power of $1$ .

$| z | = \sqrt{0 + (1 - \sin (x) - \sin (x) + \sin (x) \sin (x)) \sec^{2} (x)}$

Step 4.8.1.4.4

Raise $\sin (x)$ to the power of $1$ .

$| z | = \sqrt{0 + (1 - \sin (x) - \sin (x) + \sin (x) \sin (x)) \sec^{2} (x)}$

Step 4.8.1.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$| z | = \sqrt{0 + (1 - \sin (x) - \sin (x) + {\sin (x)}^{1 + 1}) \sec^{2} (x)}$

Step 4.8.1.4.6

Add $1$ and $1$ .

$| z | = \sqrt{0 + (1 - \sin (x) - \sin (x) + \sin^{2} (x)) \sec^{2} (x)}$

Step 4.8.2

Subtract $\sin (x)$ from $- \sin (x)$ .

$| z | = \sqrt{0 + (1 - 2 \sin (x) + \sin^{2} (x)) \sec^{2} (x)}$

Step 4.9

Apply the distributive property.

$| z | = \sqrt{0 + 1 \sec^{2} (x) - 2 \sin (x) \sec^{2} (x) + \sin^{2} (x) \sec^{2} (x)}$

Step 4.10

Simplify.

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Step 4.10.1

Multiply $\sec^{2} (x)$ by $1$ .

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

Step 4.10.2

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

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

Step 4.10.3

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

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

Step 4.10.4

One to any power is one.

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

Step 4.10.5

Combine $\sin^{2} (x)$ and $\frac{1}{\cos^{2} (x)}$ .

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

Step 4.11

Convert from $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ to $\tan^{2} (x)$ .

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

Step 4.12

Add $0$ and $\sec^{2} (x) - 2 \sin (x) \sec^{2} (x) + \tan^{2} (x)$ .

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

Step 5

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

$θ = \arctan (\frac{0}{\frac{1 - \sin (x)}{\cos (x)}})$

Step 6

Substitute the values of $θ = \arctan (\frac{0}{\frac{1 - \sin (x)}{\cos (x)}})$ and $| z | = \sqrt{\sec^{2} (x) - 2 \sin (x) \sec^{2} (x) + \tan^{2} (x)}$ .

$\sqrt{\sec^{2} (x) - 2 \sin (x) \sec^{2} (x) + \tan^{2} (x)} (\cos (\arctan (\frac{0}{\frac{1 - \sin (x)}{\cos (x)}})) + i \sin (\arctan (\frac{0}{\frac{1 - \sin (x)}{\cos (x)}})))$