Trigonometry Examples

1 - 2 \cos^{2} (x) + \cos^{4} (x)

$1 - 2 \cos^{2} (x) + \cos^{4} (x)$

Step 1

This is the trigonometric form of a complex number where

| z |

$| z |$ is the modulus and

θ

$θ$ is the angle created on the complex plane.

z = a + b i = | z | (\cos (θ) + i \sin (θ))

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 2

The modulus of a complex number is the distance from the origin on the complex plane.

| z | = \sqrt{a^{2} + b^{2}}

$| z | = \sqrt{a^{2} + b^{2}}$ where

z = a + b i

$z = a + b i$

Step 3

Substitute the actual values of

a = 1

$a = 1$ and

b = - 2 \cos^{2} (x)

$b = - 2 \cos^{2} (x)$ .

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

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

Step 4

Find

| z |

$| z |$ .

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Step 4.1

Apply the product rule to

- 2 \cos^{2} (x)

$- 2 \cos^{2} (x)$ .

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

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

Step 4.2

Raise

- 2

$- 2$ to the power of

2

$2$ .

| z | = \sqrt{4 {(\cos^{2} (x))}^{2} + 1^{2}}

$| z | = \sqrt{4 {(\cos^{2} (x))}^{2} + 1^{2}}$

Step 4.3

Multiply the exponents in

{(\cos^{2} (x))}^{2}

${(\cos^{2} (x))}^{2}$ .

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Step 4.3.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

| z | = \sqrt{4 {\cos (x)}^{2 \cdot 2} + 1^{2}}

$| z | = \sqrt{4 {\cos (x)}^{2 \cdot 2} + 1^{2}}$

Step 4.3.2

Multiply

2

$2$ by

2

$2$ .

| z | = \sqrt{4 \cos^{4} (x) + 1^{2}}

$| z | = \sqrt{4 \cos^{4} (x) + 1^{2}}$

| z | = \sqrt{4 \cos^{4} (x) + 1^{2}}

$| z | = \sqrt{4 \cos^{4} (x) + 1^{2}}$

Step 4.4

One to any power is one.

| z | = \sqrt{4 \cos^{4} (x) + 1}

$| z | = \sqrt{4 \cos^{4} (x) + 1}$

| z | = \sqrt{4 \cos^{4} (x) + 1}

$| z | = \sqrt{4 \cos^{4} (x) + 1}$

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{- 2 \cos^{2} (x)}{1})

$θ = \arctan (\frac{- 2 \cos^{2} (x)}{1})$

Step 6

Substitute the values of

θ = \arctan (\frac{- 2 \cos^{2} (x)}{1})

$θ = \arctan (\frac{- 2 \cos^{2} (x)}{1})$ and

| z | = \sqrt{4 \cos^{4} (x) + 1}

$| z | = \sqrt{4 \cos^{4} (x) + 1}$ .

\sqrt{4 \cos^{4} (x) + 1} (\cos (\arctan (\frac{- 2 \cos^{2} (x)}{1})) + i \sin (\arctan (\frac{- 2 \cos^{2} (x)}{1})))

$\sqrt{4 \cos^{4} (x) + 1} (\cos (\arctan (\frac{- 2 \cos^{2} (x)}{1})) + i \sin (\arctan (\frac{- 2 \cos^{2} (x)}{1})))$