Trigonometry Examples

$7 i^{243} + 10 i^{986}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $i^{243}$ as ${(i^{4})}^{60} (i^{2} \cdot i)$ .

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Step 1.1.1

Factor out $i^{240}$ .

$7 (i^{240} i^{3}) + 10 i^{986}$

Step 1.1.2

Rewrite $i^{240}$ as ${(i^{4})}^{60}$ .

$7 ({(i^{4})}^{60} i^{3}) + 10 i^{986}$

Step 1.1.3

Factor out $i^{2}$ .

$7 ({(i^{4})}^{60} (i^{2} \cdot i)) + 10 i^{986}$

Step 1.2

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

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Step 1.2.1

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

$7 ({({(i^{2})}^{2})}^{60} (i^{2} \cdot i)) + 10 i^{986}$

Step 1.2.2

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

$7 ({({(- 1)}^{2})}^{60} (i^{2} \cdot i)) + 10 i^{986}$

Step 1.2.3

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

$7 (1^{60} (i^{2} \cdot i)) + 10 i^{986}$

Step 1.3

One to any power is one.

$7 (1 (i^{2} \cdot i)) + 10 i^{986}$

Step 1.4

Multiply $i^{2} \cdot i$ by $1$ .

$7 (i^{2} \cdot i) + 10 i^{986}$

Step 1.5

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

$7 (- 1 \cdot i) + 10 i^{986}$

Step 1.6

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

$7 (- i) + 10 i^{986}$

Step 1.7

Multiply $- 1$ by $7$ .

$- 7 i + 10 i^{986}$

Step 1.8

Rewrite $i^{986}$ as ${(i^{4})}^{246} i^{2}$ .

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Step 1.8.1

Factor out $i^{984}$ .

$- 7 i + 10 (i^{984} i^{2})$

Step 1.8.2

Rewrite $i^{984}$ as ${(i^{4})}^{246}$ .

$- 7 i + 10 ({(i^{4})}^{246} i^{2})$

Step 1.9

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

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Step 1.9.1

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

$- 7 i + 10 ({({(i^{2})}^{2})}^{246} i^{2})$

Step 1.9.2

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

$- 7 i + 10 ({({(- 1)}^{2})}^{246} i^{2})$

Step 1.9.3

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

$- 7 i + 10 (1^{246} i^{2})$

Step 1.10

One to any power is one.

$- 7 i + 10 (1 i^{2})$

Step 1.11

Multiply $i^{2}$ by $1$ .

$- 7 i + 10 i^{2}$

Step 1.12

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

$- 7 i + 10 \cdot - 1$

Step 1.13

Multiply $10$ by $- 1$ .

$- 7 i - 10$

Step 2

Reorder $- 7 i$ and $- 10$ .

$- 10 - 7 i$

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 $a = - 10$ and $b = - 7$ .

$| z | = \sqrt{{(- 7)}^{2} + {(- 10)}^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

Raise $- 7$ to the power of $2$ .

$| z | = \sqrt{49 + {(- 10)}^{2}}$

Step 6.2

Raise $- 10$ to the power of $2$ .

$| z | = \sqrt{49 + 100}$

Step 6.3

Add $49$ and $100$ .

$| z | = \sqrt{149}$

Step 7

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

$θ = \arctan (\frac{- 7}{- 10})$

Step 8

Since inverse tangent of $\frac{- 7}{- 10}$ produces an angle in the third quadrant, the value of the angle is $3.75231861$ .

$θ = 3.75231861$

Step 9

Substitute the values of $θ = 3.75231861$ and $| z | = \sqrt{149}$ .

$\sqrt{149} (\cos (3.75231861) + i \sin (3.75231861))$