Pre-Algebra Examples

$8 (\cos (40) + i \sin (40) \cdot (4 \cos (135) + i \sin (135)))$

Step 1

Evaluate $\cos (40)$ .

$8 (0.76604444 + i \sin (40) \cdot (4 \cos (135) + i \sin (135)))$

Step 2

Evaluate $\sin (40)$ .

$8 (0.76604444 + i \cdot 0.6427876 \cdot (4 \cos (135) + i \sin (135)))$

Step 3

Move $0.6427876$ to the left of $i$ .

$8 (0.76604444 + 0.6427876 i \cdot (4 \cos (135) + i \sin (135)))$

Step 4

Simplify each term.

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Step 4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$8 (0.76604444 + 0.6427876 i \cdot (4 (- \cos (45)) + i \sin (135)))$

Step 4.2

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$8 (0.76604444 + 0.6427876 i \cdot (4 (- \frac{\sqrt{2}}{2}) + i \sin (135)))$

Step 4.3

Cancel the common factor of $2$ .

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Step 4.3.1

Move the leading negative in $- \frac{\sqrt{2}}{2}$ into the numerator.

$8 (0.76604444 + 0.6427876 i \cdot (4 \frac{- \sqrt{2}}{2} + i \sin (135)))$

Step 4.3.2

Factor $2$ out of $4$ .

$8 (0.76604444 + 0.6427876 i \cdot (2 (2) \frac{- \sqrt{2}}{2} + i \sin (135)))$

Step 4.3.3

Cancel the common factor.

$8 (0.76604444 + 0.6427876 i \cdot (2 \cdot 2 \frac{- \sqrt{2}}{2} + i \sin (135)))$

Step 4.3.4

Rewrite the expression.

$8 (0.76604444 + 0.6427876 i \cdot (2 (- \sqrt{2}) + i \sin (135)))$

Step 4.4

Multiply $- 1$ by $2$ .

$8 (0.76604444 + 0.6427876 i \cdot (- 2 \sqrt{2} + i \sin (135)))$

Step 4.5

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$8 (0.76604444 + 0.6427876 i \cdot (- 2 \sqrt{2} + i \sin (45)))$

Step 4.6

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$8 (0.76604444 + 0.6427876 i \cdot (- 2 \sqrt{2} + i \frac{\sqrt{2}}{2}))$

Step 4.7

Combine $i$ and $\frac{\sqrt{2}}{2}$ .

$8 (0.76604444 + 0.6427876 i \cdot (- 2 \sqrt{2} + \frac{i \sqrt{2}}{2}))$

$8 (0.76604444 + 0.6427876 i (- 2 \sqrt{2} + \frac{i \sqrt{2}}{2}))$

Step 5

Apply the distributive property.

$8 (0.76604444 + 0.6427876 i (- 2 \sqrt{2}) + 0.6427876 i \frac{i \sqrt{2}}{2})$

Step 6

Multiply $0.6427876 i (- 2 \sqrt{2})$ .

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Step 6.1

Multiply $- 2$ by $0.6427876$ .

$8 (0.76604444 - 1.28557521 i \sqrt{2} + 0.6427876 i \frac{i \sqrt{2}}{2})$

Step 6.2

Multiply $\sqrt{2}$ by $- 1.28557521$ .

$8 (0.76604444 - 1.81807791 i + 0.6427876 i \frac{i \sqrt{2}}{2})$

Step 7

Multiply $0.6427876 i \frac{i \sqrt{2}}{2}$ .

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Step 7.1

Combine $\frac{i \sqrt{2}}{2}$ and $0.6427876$ .

$8 (0.76604444 - 1.81807791 i + \frac{i \sqrt{2} \cdot 0.6427876}{2} i)$

Step 7.2

Multiply $0.6427876$ by $\sqrt{2}$ .

$8 (0.76604444 - 1.81807791 i + \frac{i \cdot 0.90903895}{2} i)$

Step 7.3

Combine $\frac{i \cdot 0.90903895}{2}$ and $i$ .

$8 (0.76604444 - 1.81807791 i + \frac{i \cdot 0.90903895 i}{2})$

Step 7.4

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

$8 (0.76604444 - 1.81807791 i + \frac{0.90903895 (i^{1} i)}{2})$

Step 7.5

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

$8 (0.76604444 - 1.81807791 i + \frac{0.90903895 (i^{1} i^{1})}{2})$

Step 7.6

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

$8 (0.76604444 - 1.81807791 i + \frac{0.90903895 i^{1 + 1}}{2})$

Step 7.7

Add $1$ and $1$ .

$8 (0.76604444 - 1.81807791 i + \frac{0.90903895 i^{2}}{2})$

Step 8

Simplify each term.

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Step 8.1

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

$8 (0.76604444 - 1.81807791 i + \frac{0.90903895 \cdot - 1}{2})$

Step 8.2

Multiply $0.90903895$ by $- 1$ .

$8 (0.76604444 - 1.81807791 i + \frac{- 0.90903895}{2})$

Step 8.3

Divide $- 0.90903895$ by $2$ .

$8 (0.76604444 - 1.81807791 i - 0.45451947)$

Step 9

Subtract $0.45451947$ from $0.76604444$ .

$8 (0.31152496 - 1.81807791 i)$