Precalculus Examples

$5 (\cos (25 °) + i \sin (25 °)) \cdot 2 (\cos (80 °) + i \sin (80 °))$

Step 1

Simplify terms.

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

Evaluate $\cos (25 °)$ .

$5 (0.90630778 + i \sin (25 °)) \cdot 2 (\cos (80 °) + i \sin (80 °))$

Step 1.1.2

Evaluate $\sin (25 °)$ .

$5 (0.90630778 + i \cdot 0.42261826) \cdot 2 (\cos (80 °) + i \sin (80 °))$

Step 1.1.3

Move $0.42261826$ to the left of $i$ .

$5 (0.90630778 + 0.42261826 i) \cdot 2 (\cos (80 °) + i \sin (80 °))$

Step 1.2

Simplify by multiplying through.

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$(5 \cdot 0.90630778 + 5 (0.42261826 i)) \cdot 2 (\cos (80 °) + i \sin (80 °))$

Step 1.2.2

Multiply.

Tap for more steps...

Step 1.2.2.1

Multiply $5$ by $0.90630778$ .

$(4.53153893 + 5 (0.42261826 i)) \cdot 2 (\cos (80 °) + i \sin (80 °))$

Step 1.2.2.2

Multiply $0.42261826$ by $5$ .

$(4.53153893 + 2.1130913 i) \cdot 2 (\cos (80 °) + i \sin (80 °))$

Step 1.2.3

Apply the distributive property.

$(4.53153893 \cdot 2 + 2.1130913 i \cdot 2) (\cos (80 °) + i \sin (80 °))$

Step 1.2.4

Multiply.

Tap for more steps...

Step 1.2.4.1

Multiply $4.53153893$ by $2$ .

$(9.06307787 + 2.1130913 i \cdot 2) (\cos (80 °) + i \sin (80 °))$

Step 1.2.4.2

Multiply $2$ by $2.1130913$ .

$(9.06307787 + 4.22618261 i) (\cos (80 °) + i \sin (80 °))$

Step 1.3

Simplify each term.

Tap for more steps...

Step 1.3.1

Evaluate $\cos (80 °)$ .

$(9.06307787 + 4.22618261 i) (0.17364817 + i \sin (80 °))$

Step 1.3.2

Evaluate $\sin (80 °)$ .

$(9.06307787 + 4.22618261 i) (0.17364817 + i \cdot 0.98480775)$

Step 1.3.3

Move $0.98480775$ to the left of $i$ .

$(9.06307787 + 4.22618261 i) (0.17364817 + 0.98480775 i)$

Step 2

Expand $(9.06307787 + 4.22618261 i) (0.17364817 + 0.98480775 i)$ using the FOIL Method.

Tap for more steps...

Step 2.1

Apply the distributive property.

$9.06307787 (0.17364817 + 0.98480775 i) + 4.22618261 i (0.17364817 + 0.98480775 i)$

Step 2.2

Apply the distributive property.

$9.06307787 \cdot 0.17364817 + 9.06307787 (0.98480775 i) + 4.22618261 i (0.17364817 + 0.98480775 i)$

Step 2.3

Apply the distributive property.

$9.06307787 \cdot 0.17364817 + 9.06307787 (0.98480775 i) + 4.22618261 i \cdot 0.17364817 + 4.22618261 i (0.98480775 i)$

Step 3

Simplify and combine like terms.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

Multiply $9.06307787$ by $0.17364817$ .

$1.57378695 + 9.06307787 (0.98480775 i) + 4.22618261 i \cdot 0.17364817 + 4.22618261 i (0.98480775 i)$

Step 3.1.2

Multiply $0.98480775$ by $9.06307787$ .

$1.57378695 + 8.92538935 i + 4.22618261 i \cdot 0.17364817 + 4.22618261 i (0.98480775 i)$

Step 3.1.3

Multiply $0.17364817$ by $4.22618261$ .

$1.57378695 + 8.92538935 i + 0.73386891 i + 4.22618261 i (0.98480775 i)$

Step 3.1.4

Multiply $4.22618261 i (0.98480775 i)$ .

Tap for more steps...

Step 3.1.4.1

Multiply $0.98480775$ by $4.22618261$ .

$1.57378695 + 8.92538935 i + 0.73386891 i + 4.1619774 i i$

Step 3.1.4.2

Raise $i$ to the power of $1$ .

$1.57378695 + 8.92538935 i + 0.73386891 i + 4.1619774 (i^{1} i)$

Step 3.1.4.3

Raise $i$ to the power of $1$ .

$1.57378695 + 8.92538935 i + 0.73386891 i + 4.1619774 (i^{1} i^{1})$

Step 3.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1.57378695 + 8.92538935 i + 0.73386891 i + 4.1619774 i^{1 + 1}$

Step 3.1.4.5

Add $1$ and $1$ .

$1.57378695 + 8.92538935 i + 0.73386891 i + 4.1619774 i^{2}$

Step 3.1.5

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

$1.57378695 + 8.92538935 i + 0.73386891 i + 4.1619774 \cdot - 1$

Step 3.1.6

Multiply $4.1619774$ by $- 1$ .

$1.57378695 + 8.92538935 i + 0.73386891 i - 4.1619774$

Step 3.2

Subtract $4.1619774$ from $1.57378695$ .

$- 2.58819045 + 8.92538935 i + 0.73386891 i$

Step 3.3

Add $8.92538935 i$ and $0.73386891 i$ .

$- 2.58819045 + 9.65925826 i$

Step 4

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 5

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 6

Substitute the actual values of $a = - 2.58819045$ and $b = 9.65925826$ .

$| z | = \sqrt{{9.65925826}^{2} + {(- 2.58819045)}^{2}}$

Step 7

Find $| z |$ .

Tap for more steps...

Step 7.1

Raise $9.65925826$ to the power of $2$ .

$| z | = \sqrt{93.30127018 + {(- 2.58819045)}^{2}}$

Step 7.2

Raise $- 2.58819045$ to the power of $2$ .

$| z | = \sqrt{93.30127018 + 6.69872981}$

Step 7.3

Add $93.30127018$ and $6.69872981$ .

$| z | = \sqrt{100}$

Step 7.4

Rewrite $100$ as $10^{2}$ .

$| z | = \sqrt{10^{2}}$

Step 7.5

Pull terms out from under the radical, assuming positive real numbers.

$| z | = 10$

Step 8

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

$θ = \arctan (\frac{9.65925826}{- 2.58819045})$

Step 9

The inverse tangent of $\frac{9.65925826}{- 2.58819045}$ is $- 1.30899693$ .

$θ = - 1.30899693$

Step 10

Substitute the values of $θ = - 1.30899693$ and $| z | = 10$ .

$10 (\cos (- 1.30899693) + i \sin (- 1.30899693))$