Trigonometry Examples

$\cos (θ) + \sin (θ) \cdot \tan (θ)$

Step 1

Simplify each term.

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Step 1.1

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

$\cos (θ) + \frac{\sin (θ) \sin (θ)}{\cos (θ)}$

Step 1.2

Multiply $\sin (θ) \frac{\sin (θ)}{\cos (θ)}$ .

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Step 1.2.1

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

$\cos (θ) + \frac{\sin^{1} (θ) \sin (θ)}{\cos (θ)}$

Step 1.2.2

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

$\cos (θ) + \frac{\sin^{1} (θ) \sin^{1} (θ)}{\cos (θ)}$

Step 1.2.3

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

$\cos (θ) + \frac{{\sin (θ)}^{1 + 1}}{\cos (θ)}$

Step 1.2.4

Add $1$ and $1$ .

$\cos (θ) + \frac{\sin^{2} (θ)}{\cos (θ)}$

Step 2

Simplify each term.

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Step 2.1

Factor $\sin (θ)$ out of $\sin^{2} (θ)$ .

$\cos (θ) + \frac{\sin (θ) \sin (θ)}{\cos (θ)}$

Step 2.2

Separate fractions.

$\cos (θ) + \frac{\sin (θ)}{1} \cdot \frac{\sin (θ)}{\cos (θ)}$

Step 2.3

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

$\cos (θ) + \frac{\sin (θ)}{1} \tan (θ)$

Step 2.4

Divide $\sin (θ)$ by $1$ .

$\cos (θ) + \sin (θ) \tan (θ)$

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 $a = \cos (θ)$ and $b = \sin (θ) \tan (θ)$ .

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

Step 6

Find $| z |$ .

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Multiply $\sin (θ) \frac{\sin^{2} (θ)}{\cos^{2} (θ)}$ .

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Step 6.3.1

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

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

Step 6.3.2

Multiply $\sin (θ)$ by $\sin^{2} (θ)$ by adding the exponents.

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Step 6.3.2.1

Multiply $\sin (θ)$ by $\sin^{2} (θ)$ .

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Step 6.3.2.1.1

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

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

Step 6.3.2.1.2

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

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

Step 6.3.2.2

Add $1$ and $2$ .

$| z | = \sqrt{\frac{\sin^{3} (θ)}{\cos^{2} (θ)} + \cos^{2} (θ)}$

Step 6.4

Simplify each term.

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Step 6.4.1

Factor $\sin^{2} (θ)$ out of $\sin^{3} (θ)$ .

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

Step 6.4.2

Multiply by $1$ .

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

Step 6.4.3

Separate fractions.

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

Step 6.4.4

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

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

Step 6.4.5

Divide $\sin (θ)$ by $1$ .

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

Step 7

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

$θ = \arctan (\frac{\sin (θ) \tan (θ)}{\cos (θ)})$

Step 8

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

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