Trigonometry Examples

$\sin^{4} (x) - \cos^{4} (x)$

Step 1

Simplify the expression.

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Step 1.1

Rewrite $\sin^{4} (x)$ as ${(\sin^{2} (x))}^{2}$ .

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

Step 1.2

Rewrite $\cos^{4} (x)$ as ${(\cos^{2} (x))}^{2}$ .

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

Step 2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \sin^{2} (x)$ and $b = \cos^{2} (x)$ .

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

Step 3

Apply pythagorean identity.

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

Step 4

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

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

Step 5

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 6

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 7

Substitute the actual values of $a = \sin^{2} (x)$ and $b = - 1 \cos^{2} (x)$ .

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

Step 8

Find $| z |$ .

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Step 8.1

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

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

Step 8.2

Apply the product rule to $- \cos^{2} (x)$ .

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Multiply the exponents in ${(\cos^{2} (x))}^{2}$ .

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Step 8.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 8.5.2

Multiply $2$ by $2$ .

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

Step 8.6

Multiply the exponents in ${(\sin^{2} (x))}^{2}$ .

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Step 8.6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 8.6.2

Multiply $2$ by $2$ .

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

Step 9

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

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

Step 10

Substitute the values of $θ = \arctan (\frac{- 1 \cos^{2} (x)}{\sin^{2} (x)})$ and $| z | = \sqrt{\cos^{4} (x) + \sin^{4} (x)}$ .

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