Trigonometry Examples

sin (5 x)

$\sin (5 x)$

Step 1

A good method to expand

sin (5 x)

$\sin (5 x)$ is by using De Moivre's theorem

(r {(cos (x) + i \cdot sin (x))}^{n} = r^{n} (cos (n x) + i \cdot sin (n x)))

$(r {(\cos (x) + i \cdot \sin (x))}^{n} = r^{n} (\cos (n x) + i \cdot \sin (n x)))$ . When

r = 1

$r = 1$ ,

cos (n x) + i \cdot sin (n x) = {(cos (x) + i \cdot sin (x))}^{n}

$\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}$ .

cos (n x) + i \cdot sin (n x) = {(cos (x) + i \cdot sin (x))}^{n}

$\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}$

Step 2

Expand the right hand side of

cos (n x) + i \cdot sin (n x) = {(cos (x) + i \cdot sin (x))}^{n}

$\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}$ using the binomial theorem.

Expand:

{(cos (x) + i \cdot sin (x))}^{5}

${(\cos (x) + i \cdot \sin (x))}^{5}$

Step 3

Use the Binomial Theorem.

{cos}^{5} (x) + 5 {cos}^{4} (x) (i sin (x)) + 10 {cos}^{3} (x) {(i sin (x))}^{2} + 10 {cos}^{2} (x) {(i sin (x))}^{3} + 5 cos (x) {(i sin (x))}^{4} + {(i sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) (i \sin (x)) + 10 \cos^{3} (x) {(i \sin (x))}^{2} + 10 \cos^{2} (x) {(i \sin (x))}^{3} + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Apply the product rule to

i sin (x)

$i \sin (x)$ .

{cos}^{5} (x) + 5 {cos}^{4} (x) i sin (x) + 10 {cos}^{3} (x) (i^{2} {sin}^{2} (x)) + 10 {cos}^{2} (x) {(i sin (x))}^{3} + 5 cos (x) {(i sin (x))}^{4} + {(i sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) + 10 \cos^{3} (x) (i^{2} \sin^{2} (x)) + 10 \cos^{2} (x) {(i \sin (x))}^{3} + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Step 4.1.2

Rewrite using the commutative property of multiplication.

{cos}^{5} (x) + 5 {cos}^{4} (x) i sin (x) + 10 \cdot i^{2} {cos}^{3} (x) {sin}^{2} (x) + 10 {cos}^{2} (x) {(i sin (x))}^{3} + 5 cos (x) {(i sin (x))}^{4} + {(i sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) + 10 \cdot i^{2} \cos^{3} (x) \sin^{2} (x) + 10 \cos^{2} (x) {(i \sin (x))}^{3} + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Step 4.1.3

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

{cos}^{5} (x) + 5 {cos}^{4} (x) i sin (x) + 10 \cdot - 1 {cos}^{3} (x) {sin}^{2} (x) + 10 {cos}^{2} (x) {(i sin (x))}^{3} + 5 cos (x) {(i sin (x))}^{4} + {(i sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) + 10 \cdot - 1 \cos^{3} (x) \sin^{2} (x) + 10 \cos^{2} (x) {(i \sin (x))}^{3} + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Step 4.1.4

Multiply

10

$10$ by

- 1

$- 1$ .

{cos}^{5} (x) + 5 {cos}^{4} (x) i sin (x) - 10 {cos}^{3} (x) {sin}^{2} (x) + 10 {cos}^{2} (x) {(i sin (x))}^{3} + 5 cos (x) {(i sin (x))}^{4} + {(i sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) + 10 \cos^{2} (x) {(i \sin (x))}^{3} + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Step 4.1.5

Apply the product rule to

i sin (x)

$i \sin (x)$ .

{cos}^{5} (x) + 5 {cos}^{4} (x) i sin (x) - 10 {cos}^{3} (x) {sin}^{2} (x) + 10 {cos}^{2} (x) (i^{3} {sin}^{3} (x)) + 5 cos (x) {(i sin (x))}^{4} + {(i sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) + 10 \cos^{2} (x) (i^{3} \sin^{3} (x)) + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Step 4.1.6

Rewrite using the commutative property of multiplication.

{cos}^{5} (x) + 5 {cos}^{4} (x) i sin (x) - 10 {cos}^{3} (x) {sin}^{2} (x) + 10 \cdot i^{3} {cos}^{2} (x) {sin}^{3} (x) + 5 cos (x) {(i sin (x))}^{4} + {(i sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) + 10 \cdot i^{3} \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Step 4.1.7

Factor out

i^{2}

$i^{2}$ .

{cos}^{5} (x) + 5 {cos}^{4} (x) i sin (x) - 10 {cos}^{3} (x) {sin}^{2} (x) + 10 \cdot (i^{2} \cdot i) {cos}^{2} (x) {sin}^{3} (x) + 5 cos (x) {(i sin (x))}^{4} + {(i sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) + 10 \cdot (i^{2} \cdot i) \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Step 4.1.8

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

{cos}^{5} (x) + 5 {cos}^{4} (x) i sin (x) - 10 {cos}^{3} (x) {sin}^{2} (x) + 10 \cdot (- 1 \cdot i) {cos}^{2} (x) {sin}^{3} (x) + 5 cos (x) {(i sin (x))}^{4} + {(i sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) + 10 \cdot (- 1 \cdot i) \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Step 4.1.9

Rewrite

- 1 i

$- 1 i$ as

- i

$- i$ .

{cos}^{5} (x) + 5 {cos}^{4} (x) i sin (x) - 10 {cos}^{3} (x) {sin}^{2} (x) + 10 \cdot (- i) {cos}^{2} (x) {sin}^{3} (x) + 5 cos (x) {(i sin (x))}^{4} + {(i sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) + 10 \cdot (- i) \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Step 4.1.10

Multiply

- 1

$- 1$ by

10

$10$ .

{cos}^{5} (x) + 5 {cos}^{4} (x) i sin (x) - 10 {cos}^{3} (x) {sin}^{2} (x) - 10 i {cos}^{2} (x) {sin}^{3} (x) + 5 cos (x) {(i sin (x))}^{4} + {(i sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) {(i \sin (x))}^{4} + {(i \sin (x))}^{5}$

Step 4.1.11

Apply the product rule to

i sin (x)

$i \sin (x)$ .

{cos}^{5} (x) + 5 {cos}^{4} (x) i sin (x) - 10 {cos}^{3} (x) {sin}^{2} (x) - 10 i {cos}^{2} (x) {sin}^{3} (x) + 5 cos (x) (i^{4} {sin}^{4} (x)) + {(i sin (x))}^{5}

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) (i^{4} \sin^{4} (x)) + {(i \sin (x))}^{5}$

Step 4.1.12

Rewrite

$i^{4}$ as

$1$ .

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Step 4.1.12.1

Rewrite

$i^{4}$ as

${(i^{2})}^{2}$ .

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) ({(i^{2})}^{2} \sin^{4} (x)) + {(i \sin (x))}^{5}$

Step 4.1.12.2

Rewrite

$i^{2}$ as

$- 1$ .

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) ({(- 1)}^{2} \sin^{4} (x)) + {(i \sin (x))}^{5}$

Step 4.1.12.3

Raise

$- 1$ to the power of

$2$ .

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) (1 \sin^{4} (x)) + {(i \sin (x))}^{5}$

Step 4.1.13

Multiply

$\sin^{4} (x)$ by

$1$ .

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + {(i \sin (x))}^{5}$

Step 4.1.14

Apply the product rule to

$i \sin (x)$ .

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + i^{5} \sin^{5} (x)$

Step 4.1.15

Factor out

$i^{4}$ .

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + i^{4} i \sin^{5} (x)$

Step 4.1.16

Rewrite

$i^{4}$ as

$1$ .

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Step 4.1.16.1

Rewrite

$i^{4}$ as

${(i^{2})}^{2}$ .

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + {(i^{2})}^{2} i \sin^{5} (x)$

Step 4.1.16.2

Rewrite

$i^{2}$ as

$- 1$ .

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + {(- 1)}^{2} i \sin^{5} (x)$

Step 4.1.16.3

Raise

$- 1$ to the power of

$2$ .

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + 1 i \sin^{5} (x)$

Step 4.1.17

Multiply

$i$ by

$1$ .

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + i \sin^{5} (x)$

Step 4.2

Reorder factors in

$\cos^{5} (x) + 5 \cos^{4} (x) i \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + i \sin^{5} (x)$ .

$\cos^{5} (x) + 5 i \cos^{4} (x) \sin (x) - 10 \cos^{3} (x) \sin^{2} (x) - 10 i \cos^{2} (x) \sin^{3} (x) + 5 \cos (x) \sin^{4} (x) + i \sin^{5} (x)$

Step 5

Take out the expressions with the imaginary part, which are equal to

$\sin (5 x)$ . Remove the imaginary number

$i$ .

$\sin (5 x) = 5 \cos^{4} (x) \sin (x) - 10 \cos^{2} (x) \sin^{3} (x) + \sin^{5} (x)$

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