Trigonometry Examples

$\frac{1 + \cos (2 y)}{\sin (2 y)}$

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 $a = \frac{1 + \cos (2 y)}{\sin (2 y)}$ and $b = 0$ .

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

Step 4

Find $| z |$ .

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Step 4.1

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

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

Step 4.2

Apply the product rule to $\frac{1 + \cos (2 y)}{\sin (2 y)}$ .

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

Step 4.3

Multiply by $1$ .

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

Step 4.4

Separate fractions.

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

Step 4.5

Convert from $\frac{1}{\sin^{2} (2 y)}$ to $\csc^{2} (2 y)$ .

$| z | = \sqrt{0 + \frac{{(1 + \cos (2 y))}^{2}}{1} \cdot \csc^{2} (2 y)}$

Step 4.6

Simplify the expression.

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Step 4.6.1

Divide ${(1 + \cos (2 y))}^{2}$ by $1$ .

$| z | = \sqrt{0 + {(1 + \cos (2 y))}^{2} \csc^{2} (2 y)}$

Step 4.6.2

Rewrite ${(1 + \cos (2 y))}^{2}$ as $(1 + \cos (2 y)) (1 + \cos (2 y))$ .

$| z | = \sqrt{0 + (1 + \cos (2 y)) (1 + \cos (2 y)) \csc^{2} (2 y)}$

Step 4.7

Expand $(1 + \cos (2 y)) (1 + \cos (2 y))$ using the FOIL Method.

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Step 4.7.1

Apply the distributive property.

$| z | = \sqrt{0 + (1 (1 + \cos (2 y)) + \cos (2 y) (1 + \cos (2 y))) \csc^{2} (2 y)}$

Step 4.7.2

Apply the distributive property.

$| z | = \sqrt{0 + (1 \cdot 1 + 1 \cos (2 y) + \cos (2 y) (1 + \cos (2 y))) \csc^{2} (2 y)}$

Step 4.7.3

Apply the distributive property.

$| z | = \sqrt{0 + (1 \cdot 1 + 1 \cos (2 y) + \cos (2 y) \cdot 1 + \cos (2 y) \cos (2 y)) \csc^{2} (2 y)}$

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 $1$ .

$| z | = \sqrt{0 + (1 + 1 \cos (2 y) + \cos (2 y) \cdot 1 + \cos (2 y) \cos (2 y)) \csc^{2} (2 y)}$

Step 4.8.1.2

Multiply $\cos (2 y)$ by $1$ .

$| z | = \sqrt{0 + (1 + \cos (2 y) + \cos (2 y) \cdot 1 + \cos (2 y) \cos (2 y)) \csc^{2} (2 y)}$

Step 4.8.1.3

Multiply $\cos (2 y)$ by $1$ .

$| z | = \sqrt{0 + (1 + \cos (2 y) + \cos (2 y) + \cos (2 y) \cos (2 y)) \csc^{2} (2 y)}$

Step 4.8.1.4

Multiply $\cos (2 y) \cos (2 y)$ .

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Step 4.8.1.4.1

Raise $\cos (2 y)$ to the power of $1$ .

$| z | = \sqrt{0 + (1 + \cos (2 y) + \cos (2 y) + \cos (2 y) \cos (2 y)) \csc^{2} (2 y)}$

Step 4.8.1.4.2

Raise $\cos (2 y)$ to the power of $1$ .

$| z | = \sqrt{0 + (1 + \cos (2 y) + \cos (2 y) + \cos (2 y) \cos (2 y)) \csc^{2} (2 y)}$

Step 4.8.1.4.3

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

$| z | = \sqrt{0 + (1 + \cos (2 y) + \cos (2 y) + {\cos (2 y)}^{1 + 1}) \csc^{2} (2 y)}$

Step 4.8.1.4.4

Add $1$ and $1$ .

$| z | = \sqrt{0 + (1 + \cos (2 y) + \cos (2 y) + \cos^{2} (2 y)) \csc^{2} (2 y)}$

Step 4.8.2

Add $\cos (2 y)$ and $\cos (2 y)$ .

$| z | = \sqrt{0 + (1 + 2 \cos (2 y) + \cos^{2} (2 y)) \csc^{2} (2 y)}$

Step 4.9

Apply the distributive property.

$| z | = \sqrt{0 + 1 \csc^{2} (2 y) + 2 \cos (2 y) \csc^{2} (2 y) + \cos^{2} (2 y) \csc^{2} (2 y)}$

Step 4.10

Simplify.

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Step 4.10.1

Multiply $\csc^{2} (2 y)$ by $1$ .

$| z | = \sqrt{0 + \csc^{2} (2 y) + 2 \cos (2 y) \csc^{2} (2 y) + \cos^{2} (2 y) \csc^{2} (2 y)}$

Step 4.10.2

Rewrite $\csc (2 y)$ in terms of sines and cosines.

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

Step 4.10.3

Apply the product rule to $\frac{1}{\sin (2 y)}$ .

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

Step 4.10.4

One to any power is one.

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

Step 4.10.5

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

$| z | = \sqrt{0 + \csc^{2} (2 y) + 2 \cos (2 y) \csc^{2} (2 y) + \frac{\cos^{2} (2 y)}{\sin^{2} (2 y)}}$

Step 4.11

Convert from $\frac{\cos^{2} (2 y)}{\sin^{2} (2 y)}$ to $\cot^{2} (2 y)$ .

$| z | = \sqrt{0 + \csc^{2} (2 y) + 2 \cos (2 y) \csc^{2} (2 y) + \cot^{2} (2 y)}$

Step 4.12

Add $0$ and $\csc^{2} (2 y) + 2 \cos (2 y) \csc^{2} (2 y) + \cot^{2} (2 y)$ .

$| z | = \sqrt{\csc^{2} (2 y) + 2 \cos (2 y) \csc^{2} (2 y) + \cot^{2} (2 y)}$

Step 4.13

Rewrite $\csc^{2} (2 y) + 2 \cos (2 y) \csc^{2} (2 y) + \cot^{2} (2 y)$ in a factored form.

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Step 4.13.1

Rewrite the middle term.

$| z | = \sqrt{\csc^{2} (2 y) + 2 \csc (2 y) \cot (2 y) + 0 + \cot^{2} (2 y)}$

Step 4.13.2

Rearrange terms.

$| z | = \sqrt{\csc^{2} (2 y) + 2 \csc (2 y) \cot (2 y) + \cot^{2} (2 y) + 0}$

Step 4.13.3

Factor first three terms by perfect square rule.

$| z | = \sqrt{{(\csc (2 y) + \cot (2 y))}^{2} + 0}$

Step 4.13.4

Rewrite ${(\csc (2 y) + \cot (2 y))}^{2}$ as $(\csc (2 y) + \cot (2 y)) (\csc (2 y) + \cot (2 y))$ .

$| z | = \sqrt{(\csc (2 y) + \cot (2 y)) (\csc (2 y) + \cot (2 y)) + 0}$

Step 4.13.5

Expand $(\csc (2 y) + \cot (2 y)) (\csc (2 y) + \cot (2 y))$ using the FOIL Method.

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Step 4.13.5.1

Apply the distributive property.

$| z | = \sqrt{\csc (2 y) (\csc (2 y) + \cot (2 y)) + \cot (2 y) (\csc (2 y) + \cot (2 y)) + 0}$

Step 4.13.5.2

Apply the distributive property.

$| z | = \sqrt{\csc (2 y) \csc (2 y) + \csc (2 y) \cot (2 y) + \cot (2 y) (\csc (2 y) + \cot (2 y)) + 0}$

Step 4.13.5.3

Apply the distributive property.

$| z | = \sqrt{\csc (2 y) \csc (2 y) + \csc (2 y) \cot (2 y) + \cot (2 y) \csc (2 y) + \cot (2 y) \cot (2 y) + 0}$

Step 4.13.6

Simplify and combine like terms.

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Step 4.13.6.1

Simplify each term.

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Step 4.13.6.1.1

Multiply $\csc (2 y) \csc (2 y)$ .

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Step 4.13.6.1.1.1

Raise $\csc (2 y)$ to the power of $1$ .

$| z | = \sqrt{\csc (2 y) \csc (2 y) + \csc (2 y) \cot (2 y) + \cot (2 y) \csc (2 y) + \cot (2 y) \cot (2 y) + 0}$

Step 4.13.6.1.1.2

Raise $\csc (2 y)$ to the power of $1$ .

$| z | = \sqrt{\csc (2 y) \csc (2 y) + \csc (2 y) \cot (2 y) + \cot (2 y) \csc (2 y) + \cot (2 y) \cot (2 y) + 0}$

Step 4.13.6.1.1.3

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

$| z | = \sqrt{{\csc (2 y)}^{1 + 1} + \csc (2 y) \cot (2 y) + \cot (2 y) \csc (2 y) + \cot (2 y) \cot (2 y) + 0}$

Step 4.13.6.1.1.4

Add $1$ and $1$ .

$| z | = \sqrt{\csc^{2} (2 y) + \csc (2 y) \cot (2 y) + \cot (2 y) \csc (2 y) + \cot (2 y) \cot (2 y) + 0}$

Step 4.13.6.1.2

Multiply $\cot (2 y) \cot (2 y)$ .

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Step 4.13.6.1.2.1

Raise $\cot (2 y)$ to the power of $1$ .

$| z | = \sqrt{\csc^{2} (2 y) + \csc (2 y) \cot (2 y) + \cot (2 y) \csc (2 y) + \cot (2 y) \cot (2 y) + 0}$

Step 4.13.6.1.2.2

Raise $\cot (2 y)$ to the power of $1$ .

$| z | = \sqrt{\csc^{2} (2 y) + \csc (2 y) \cot (2 y) + \cot (2 y) \csc (2 y) + \cot (2 y) \cot (2 y) + 0}$

Step 4.13.6.1.2.3

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

$| z | = \sqrt{\csc^{2} (2 y) + \csc (2 y) \cot (2 y) + \cot (2 y) \csc (2 y) + {\cot (2 y)}^{1 + 1} + 0}$

Step 4.13.6.1.2.4

Add $1$ and $1$ .

$| z | = \sqrt{\csc^{2} (2 y) + \csc (2 y) \cot (2 y) + \cot (2 y) \csc (2 y) + \cot^{2} (2 y) + 0}$

Step 4.13.6.2

Reorder the factors of $\csc (2 y) \cot (2 y)$ .

$| z | = \sqrt{\csc^{2} (2 y) + \cot (2 y) \csc (2 y) + \cot (2 y) \csc (2 y) + \cot^{2} (2 y) + 0}$

Step 4.13.6.3

Add $\cot (2 y) \csc (2 y)$ and $\cot (2 y) \csc (2 y)$ .

$| z | = \sqrt{\csc^{2} (2 y) + 2 \cot (2 y) \csc (2 y) + \cot^{2} (2 y) + 0}$

Step 4.13.7

Add $\csc^{2} (2 y) + 2 \cot (2 y) \csc (2 y) + \cot^{2} (2 y)$ and $0$ .

$| z | = \sqrt{\csc^{2} (2 y) + 2 \cot (2 y) \csc (2 y) + \cot^{2} (2 y)}$

Step 4.13.8

Factor using the perfect square rule.

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Step 4.13.8.1

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 \cot (2 y) \csc (2 y) = 2 \cdot \csc (2 y) \cdot \cot (2 y)$

Step 4.13.8.2

Rewrite the polynomial.

$| z | = \sqrt{\csc^{2} (2 y) + 2 \cdot \csc (2 y) \cdot \cot (2 y) + \cot^{2} (2 y)}$

Step 4.13.8.3

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = \csc (2 y)$ and $b = \cot (2 y)$ .

$| z | = \sqrt{{(\csc (2 y) + \cot (2 y))}^{2}}$

Step 4.14

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

$| z | = \csc (2 y) + \cot (2 y)$

Step 5

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

$θ = \arctan (\frac{0}{\frac{1 + \cos (2 y)}{\sin (2 y)}})$

Step 6

Substitute the values of $θ = \arctan (\frac{0}{\frac{1 + \cos (2 y)}{\sin (2 y)}})$ and $| z | = \csc (2 y) + \cot (2 y)$ .

$\csc (2 y) + \cot (2 y) (\cos (\arctan (\frac{0}{\frac{1 + \cos (2 y)}{\sin (2 y)}})) + i \sin (\arctan (\frac{0}{\frac{1 + \cos (2 y)}{\sin (2 y)}})))$