Trigonometry Examples

$\csc (θ) - \sin (θ)$ ?

Step 1

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

$\frac{1}{\sin (θ)} - \sin (θ)$

Step 2

Convert from $\frac{1}{\sin (θ)}$ to $\csc (θ)$ .

$\csc (θ) - \sin (θ)$

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 $a = \csc (θ)$ and $b = - 1 \sin (θ)$ .

$| z | = \sqrt{{(- 1 \sin (θ))}^{2} + \csc^{2} (θ)}$

Step 6

Find $| z |$ .

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Step 6.1

Rewrite $- 1 \sin (θ)$ as $- \sin (θ)$ .

$| z | = \sqrt{{(- \sin (θ))}^{2} + \csc^{2} (θ)}$

Step 6.2

Apply the product rule to $- \sin (θ)$ .

$| z | = \sqrt{{(- 1)}^{2} \sin^{2} (θ) + \csc^{2} (θ)}$

Step 6.3

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

$| z | = \sqrt{1 \sin^{2} (θ) + \csc^{2} (θ)}$

Step 6.4

Multiply $\sin^{2} (θ)$ by $1$ .

$| z | = \sqrt{\sin^{2} (θ) + \csc^{2} (θ)}$

Step 6.5

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

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

Step 6.6

Simplify the expression.

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Step 6.6.1

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

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

Step 6.6.2

One to any power is one.

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

Step 6.7

Simplify each term.

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Step 6.7.1

Rewrite $1$ as $1^{2}$ .

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

Step 6.7.2

Rewrite $\frac{1^{2}}{\sin^{2} (θ)}$ as ${(\frac{1}{\sin (θ)})}^{2}$ .

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

Step 6.7.3

Convert from $\frac{1}{\sin (θ)}$ to $\csc (θ)$ .

$| z | = \sqrt{\sin^{2} (θ) + \csc^{2} (θ)}$

Step 7

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

$θ = \arctan (\frac{- 1 \sin (θ)}{\csc (θ)})$

Step 8

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

$\sqrt{\sin^{2} (θ) + \csc^{2} (θ)} (\cos (\arctan (\frac{- 1 \sin (θ)}{\csc (θ)})) + i \sin (\arctan (\frac{- 1 \sin (θ)}{\csc (θ)})))$