Trigonometry Examples

${(3 \cos (115) + i \sin (115))}^{4}$

Step 1

Use the Binomial Theorem.

${(3 \cos (115))}^{4} + 4 {(3 \cos (115))}^{3} (i \sin (115)) + 6 {(3 \cos (115))}^{2} {(i \sin (115))}^{2} + 4 (3 \cos (115)) {(i \sin (115))}^{3} + {(i \sin (115))}^{4}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the product rule to $3 \cos (115)$ .

$3^{4} \cos^{4} (115) + 4 {(3 \cos (115))}^{3} (i \sin (115)) + 6 {(3 \cos (115))}^{2} {(i \sin (115))}^{2} + 4 (3 \cos (115)) {(i \sin (115))}^{3} + {(i \sin (115))}^{4}$

Step 2.1.2

Raise $3$ to the power of $4$ .

$81 \cos^{4} (115) + 4 {(3 \cos (115))}^{3} (i \sin (115)) + 6 {(3 \cos (115))}^{2} {(i \sin (115))}^{2} + 4 (3 \cos (115)) {(i \sin (115))}^{3} + {(i \sin (115))}^{4}$

Step 2.1.3

Apply the product rule to $3 \cos (115)$ .

$81 \cos^{4} (115) + 4 (3^{3} \cos^{3} (115)) (i \sin (115)) + 6 {(3 \cos (115))}^{2} {(i \sin (115))}^{2} + 4 (3 \cos (115)) {(i \sin (115))}^{3} + {(i \sin (115))}^{4}$

Step 2.1.4

Raise $3$ to the power of $3$ .

$81 \cos^{4} (115) + 4 (27 \cos^{3} (115)) (i \sin (115)) + 6 {(3 \cos (115))}^{2} {(i \sin (115))}^{2} + 4 (3 \cos (115)) {(i \sin (115))}^{3} + {(i \sin (115))}^{4}$

Step 2.1.5

Multiply $27$ by $4$ .

$81 \cos^{4} (115) + 108 \cos^{3} (115) (i \sin (115)) + 6 {(3 \cos (115))}^{2} {(i \sin (115))}^{2} + 4 (3 \cos (115)) {(i \sin (115))}^{3} + {(i \sin (115))}^{4}$

Step 2.1.6

Apply the product rule to $3 \cos (115)$ .

$81 \cos^{4} (115) + 108 \cos^{3} (115) i \sin (115) + 6 (3^{2} \cos^{2} (115)) {(i \sin (115))}^{2} + 4 (3 \cos (115)) {(i \sin (115))}^{3} + {(i \sin (115))}^{4}$

Step 2.1.7

Raise $3$ to the power of $2$ .

$81 \cos^{4} (115) + 108 \cos^{3} (115) i \sin (115) + 6 (9 \cos^{2} (115)) {(i \sin (115))}^{2} + 4 (3 \cos (115)) {(i \sin (115))}^{3} + {(i \sin (115))}^{4}$

Step 2.1.8

Multiply $9$ by $6$ .

$81 \cos^{4} (115) + 108 \cos^{3} (115) i \sin (115) + 54 \cos^{2} (115) {(i \sin (115))}^{2} + 4 (3 \cos (115)) {(i \sin (115))}^{3} + {(i \sin (115))}^{4}$

Step 2.1.9

Apply the product rule to $i \sin (115)$ .

$81 \cos^{4} (115) + 108 \cos^{3} (115) i \sin (115) + 54 \cos^{2} (115) (i^{2} \sin^{2} (115)) + 4 (3 \cos (115)) {(i \sin (115))}^{3} + {(i \sin (115))}^{4}$

Step 2.1.10

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

$81 \cos^{4} (115) + 108 \cos^{3} (115) i \sin (115) + 54 \cos^{2} (115) (- 1 \sin^{2} (115)) + 4 (3 \cos (115)) {(i \sin (115))}^{3} + {(i \sin (115))}^{4}$

Step 2.1.11

Rewrite $- 1 \sin^{2} (115)$ as $- \sin^{2} (115)$ .

$81 \cos^{4} (115) + 108 \cos^{3} (115) i \sin (115) + 54 \cos^{2} (115) (- \sin^{2} (115)) + 4 (3 \cos (115)) {(i \sin (115))}^{3} + {(i \sin (115))}^{4}$

Step 2.1.12

Multiply $- 1$ by $54$ .

$81 \cos^{4} (115) + 108 \cos^{3} (115) i \sin (115) - 54 \cos^{2} (115) \sin^{2} (115) + 4 (3 \cos (115)) {(i \sin (115))}^{3} + {(i \sin (115))}^{4}$

Step 2.1.13

Multiply $3$ by $4$ .

$81 \cos^{4} (115) + 108 \cos^{3} (115) i \sin (115) - 54 \cos^{2} (115) \sin^{2} (115) + 12 \cos (115) {(i \sin (115))}^{3} + {(i \sin (115))}^{4}$

Step 2.1.14

Apply the product rule to $i \sin (115)$ .

$81 \cos^{4} (115) + 108 \cos^{3} (115) i \sin (115) - 54 \cos^{2} (115) \sin^{2} (115) + 12 \cos (115) (i^{3} \sin^{3} (115)) + {(i \sin (115))}^{4}$

Step 2.1.15

Factor out $i^{2}$ .

$81 \cos^{4} (115) + 108 \cos^{3} (115) i \sin (115) - 54 \cos^{2} (115) \sin^{2} (115) + 12 \cos (115) (i^{2} \cdot i \sin^{3} (115)) + {(i \sin (115))}^{4}$

Step 2.1.16

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

$81 \cos^{4} (115) + 108 \cos^{3} (115) i \sin (115) - 54 \cos^{2} (115) \sin^{2} (115) + 12 \cos (115) (- 1 \cdot i \sin^{3} (115)) + {(i \sin (115))}^{4}$

Step 2.1.17

Rewrite $- 1 i$ as $- i$ .

$81 \cos^{4} (115) + 108 \cos^{3} (115) i \sin (115) - 54 \cos^{2} (115) \sin^{2} (115) + 12 \cos (115) (- i \sin^{3} (115)) + {(i \sin (115))}^{4}$

Step 2.1.18

Multiply $- 1$ by $12$ .

$81 \cos^{4} (115) + 108 \cos^{3} (115) i \sin (115) - 54 \cos^{2} (115) \sin^{2} (115) - 12 \cos (115) (i \sin^{3} (115)) + {(i \sin (115))}^{4}$

Step 2.1.19

Apply the product rule to $i \sin (115)$ .

$81 \cos^{4} (115) + 108 \cos^{3} (115) i \sin (115) - 54 \cos^{2} (115) \sin^{2} (115) - 12 \cos (115) i \sin^{3} (115) + i^{4} \sin^{4} (115)$

Step 2.1.20

Rewrite $i^{4}$ as $1$ .

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Step 2.1.20.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$81 \cos^{4} (115) + 108 \cos^{3} (115) i \sin (115) - 54 \cos^{2} (115) \sin^{2} (115) - 12 \cos (115) i \sin^{3} (115) + {(i^{2})}^{2} \sin^{4} (115)$

Step 2.1.20.2

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

$81 \cos^{4} (115) + 108 \cos^{3} (115) i \sin (115) - 54 \cos^{2} (115) \sin^{2} (115) - 12 \cos (115) i \sin^{3} (115) + {(- 1)}^{2} \sin^{4} (115)$

Step 2.1.20.3

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

$81 \cos^{4} (115) + 108 \cos^{3} (115) i \sin (115) - 54 \cos^{2} (115) \sin^{2} (115) - 12 \cos (115) i \sin^{3} (115) + 1 \sin^{4} (115)$

Step 2.1.21

Multiply $\sin^{4} (115)$ by $1$ .

$81 \cos^{4} (115) + 108 \cos^{3} (115) i \sin (115) - 54 \cos^{2} (115) \sin^{2} (115) - 12 \cos (115) i \sin^{3} (115) + \sin^{4} (115)$

Step 2.2

Simplify the expression.

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Step 2.2.1

Reorder factors in $81 \cos^{4} (115) + 108 \cos^{3} (115) i \sin (115) - 54 \cos^{2} (115) \sin^{2} (115) - 12 \cos (115) i \sin^{3} (115) + \sin^{4} (115)$ .

$81 \cos^{4} (115) + 108 i \cos^{3} (115) \sin (115) - 54 \cos^{2} (115) \sin^{2} (115) - 12 i \cos (115) \sin^{3} (115) + \sin^{4} (115)$

Step 2.2.2

Reorder $81 \cos^{4} (115) + 108 i \cos^{3} (115) \sin (115)$ and $- 54 \cos^{2} (115) \sin^{2} (115)$ .

$- 54 \cos^{2} (115) \sin^{2} (115) + 81 \cos^{4} (115) + 108 i \cos^{3} (115) \sin (115) - 12 i \cos (115) \sin^{3} (115) + \sin^{4} (115)$

Step 2.2.3

Reorder $- 54 \cos^{2} (115) \sin^{2} (115) + 81 \cos^{4} (115) + 108 i \cos^{3} (115) \sin (115) - 12 i \cos (115) \sin^{3} (115)$ and $\sin^{4} (115)$ .

$\sin^{4} (115) - 54 \cos^{2} (115) \sin^{2} (115) + 81 \cos^{4} (115) + 108 i \cos^{3} (115) \sin (115) - 12 i \cos (115) \sin^{3} (115)$