Precalculus Examples

$5 (\cos (15 °) + i \sin (15 °)) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

The exact value of $\cos (15 °)$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

Tap for more steps...

Step 1.1.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$5 (\cos (45 - 30) + i \sin (15 °)) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.1.2

Separate negation.

$5 (\cos (45 - (30)) + i \sin (15 °)) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.1.3

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

$5 (\cos (45) \cos (30) + \sin (45) \sin (30) + i \sin (15 °)) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.1.4

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

$5 (\frac{\sqrt{2}}{2} \cos (30) + \sin (45) \sin (30) + i \sin (15 °)) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.1.5

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$5 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30) + i \sin (15 °)) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.1.6

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

$5 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (30) + i \sin (15 °)) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.1.7

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$5 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (15 °)) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.1.8

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 1.1.8.1

Simplify each term.

Tap for more steps...

Step 1.1.8.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

Tap for more steps...

Step 1.1.8.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$5 (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (15 °)) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.1.8.1.1.2

Combine using the product rule for radicals.

$5 (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (15 °)) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.1.8.1.1.3

Multiply $2$ by $3$ .

$5 (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (15 °)) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.1.8.1.1.4

Multiply $2$ by $2$ .

$5 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (15 °)) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.1.8.1.2

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 1.1.8.1.2.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{1}{2}$ .

$5 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2} + i \sin (15 °)) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.1.8.1.2.2

Multiply $2$ by $2$ .

$5 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} + i \sin (15 °)) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.1.8.2

Combine the numerators over the common denominator.

$5 (\frac{\sqrt{6} + \sqrt{2}}{4} + i \sin (15 °)) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.2

The exact value of $\sin (15 °)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

Tap for more steps...

Step 1.2.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$5 (\frac{\sqrt{6} + \sqrt{2}}{4} + i \sin (45 - 30)) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.2.2

Separate negation.

$5 (\frac{\sqrt{6} + \sqrt{2}}{4} + i \sin (45 - (30))) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.2.3

Apply the difference of angles identity.

$5 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\sin (45) \cos (30) - \cos (45) \sin (30))) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.2.4

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

$5 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30))) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.2.5

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$5 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30))) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.2.6

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

$5 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30))) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.2.7

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$5 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.2.8

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 1.2.8.1

Simplify each term.

Tap for more steps...

Step 1.2.8.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

Tap for more steps...

Step 1.2.8.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$5 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.2.8.1.1.2

Combine using the product rule for radicals.

$5 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.2.8.1.1.3

Multiply $2$ by $3$ .

$5 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.2.8.1.1.4

Multiply $2$ by $2$ .

$5 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.2.8.1.2

Multiply $- \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 1.2.8.1.2.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{2}}{2}$ .

$5 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2})) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.2.8.1.2.2

Multiply $2$ by $2$ .

$5 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.2.8.2

Combine the numerators over the common denominator.

$5 (\frac{\sqrt{6} + \sqrt{2}}{4} + i \frac{\sqrt{6} - \sqrt{2}}{4}) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 1.3

Combine $i$ and $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

$5 (\frac{\sqrt{6} + \sqrt{2}}{4} + \frac{i (\sqrt{6} - \sqrt{2})}{4}) \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 2

Combine fractions.

Tap for more steps...

Step 2.1

Combine the numerators over the common denominator.

$5 \frac{\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}}{4} \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 2.2

Combine $5$ and $\frac{\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}}{4}$ .

$\frac{5 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4} \cdot 3 (\cos (70 °) + i \sin (70 °))$

Step 3

Multiply $\frac{5 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4} \cdot 3$ .

Tap for more steps...

Step 3.1

Combine $\frac{5 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$ and $3$ .

$\frac{5 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}) \cdot 3}{4} (\cos (70 °) + i \sin (70 °))$

Step 3.2

Multiply $3$ by $5$ .

$\frac{15 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4} (\cos (70 °) + i \sin (70 °))$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Evaluate $\cos (70 °)$ .

$\frac{15 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4} (0.34202014 + i \sin (70 °))$

Step 4.2

Evaluate $\sin (70 °)$ .

$\frac{15 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4} (0.34202014 + i \cdot 0.93969262)$

Step 4.3

Move $0.93969262$ to the left of $i$ .

$\frac{15 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4} (0.34202014 + 0.93969262 i)$

Step 5

Apply the distributive property.

$\frac{15 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4} \cdot 0.34202014 + \frac{15 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4} (0.93969262 i)$

Step 6

Multiply $\frac{15 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4} \cdot 0.34202014$ .

Tap for more steps...

Step 6.1

Combine $\frac{15 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$ and $0.34202014$ .

$\frac{15 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}) \cdot 0.34202014}{4} + \frac{15 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4} (0.93969262 i)$

Step 6.2

Multiply $0.34202014$ by $15$ .

$\frac{5.13030214 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4} + \frac{15 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4} (0.93969262 i)$

Step 7

Multiply $\frac{15 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4} (0.93969262 i)$ .

Tap for more steps...

Step 7.1

Combine $0.93969262$ and $\frac{15 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$ .

$\frac{5.13030214 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4} + \frac{0.93969262 (15 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}))}{4} i$

Step 7.2

Multiply $15$ by $0.93969262$ .

$\frac{5.13030214 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4} + \frac{14.09538931 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4} i$

Step 7.3

Combine $\frac{14.09538931 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$ and $i$ .

$\frac{5.13030214 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4} + \frac{14.09538931 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}) i}{4}$

Step 8

Combine the numerators over the common denominator.

$\frac{5.13030214 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}) + 14.09538931 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}) i}{4}$

Step 9

Simplify each term.

Tap for more steps...

Step 9.1

Apply the distributive property.

$\frac{5.13030214 \sqrt{6} + 5.13030214 \sqrt{2} + 5.13030214 (i \sqrt{6}) + 5.13030214 (- i \sqrt{2}) + 14.09538931 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}) i}{4}$

Step 9.2

Simplify.

Tap for more steps...

Step 9.2.1

Multiply $5.13030214$ by $\sqrt{6}$ .

$\frac{12.56662249 + 5.13030214 \sqrt{2} + 5.13030214 (i \sqrt{6}) + 5.13030214 (- i \sqrt{2}) + 14.09538931 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}) i}{4}$

Step 9.2.2

Multiply $5.13030214$ by $\sqrt{2}$ .

$\frac{12.56662249 + 7.25534287 + 5.13030214 (i \sqrt{6}) + 5.13030214 (- i \sqrt{2}) + 14.09538931 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}) i}{4}$

Step 9.2.3

Multiply $\sqrt{6}$ by $5.13030214$ .

$\frac{12.56662249 + 7.25534287 + 12.56662249 i + 5.13030214 (- i \sqrt{2}) + 14.09538931 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}) i}{4}$

Step 9.2.4

Multiply $5.13030214 (- i \sqrt{2})$ .

Tap for more steps...

Step 9.2.4.1

Multiply $- 1$ by $5.13030214$ .

$\frac{12.56662249 + 7.25534287 + 12.56662249 i - 5.13030214 (i \sqrt{2}) + 14.09538931 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}) i}{4}$

Step 9.2.4.2

Multiply $\sqrt{2}$ by $- 5.13030214$ .

$\frac{12.56662249 + 7.25534287 + 12.56662249 i - 7.25534287 i + 14.09538931 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}) i}{4}$

Step 9.3

Add $12.56662249$ and $7.25534287$ .

$\frac{19.82196537 + 12.56662249 i - 7.25534287 i + 14.09538931 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}) i}{4}$

Step 9.4

Subtract $7.25534287 i$ from $12.56662249 i$ .

$\frac{19.82196537 + 5.31127961 i + 14.09538931 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}) i}{4}$

Step 9.5

Apply the distributive property.

$\frac{19.82196537 + 5.31127961 i + (14.09538931 \sqrt{6} + 14.09538931 \sqrt{2} + 14.09538931 (i \sqrt{6}) + 14.09538931 (- i \sqrt{2})) i}{4}$

Step 9.6

Simplify.

Tap for more steps...

Step 9.6.1

Multiply $14.09538931$ by $\sqrt{6}$ .

$\frac{19.82196537 + 5.31127961 i + (34.52651153 + 14.09538931 \sqrt{2} + 14.09538931 (i \sqrt{6}) + 14.09538931 (- i \sqrt{2})) i}{4}$

Step 9.6.2

Multiply $14.09538931$ by $\sqrt{2}$ .

$\frac{19.82196537 + 5.31127961 i + (34.52651153 + 19.93389073 + 14.09538931 (i \sqrt{6}) + 14.09538931 (- i \sqrt{2})) i}{4}$

Step 9.6.3

Multiply $\sqrt{6}$ by $14.09538931$ .

$\frac{19.82196537 + 5.31127961 i + (34.52651153 + 19.93389073 + 34.52651153 i + 14.09538931 (- i \sqrt{2})) i}{4}$

Step 9.6.4

Multiply $14.09538931 (- i \sqrt{2})$ .

Tap for more steps...

Step 9.6.4.1

Multiply $- 1$ by $14.09538931$ .

$\frac{19.82196537 + 5.31127961 i + (34.52651153 + 19.93389073 + 34.52651153 i - 14.09538931 (i \sqrt{2})) i}{4}$

Step 9.6.4.2

Multiply $\sqrt{2}$ by $- 14.09538931$ .

$\frac{19.82196537 + 5.31127961 i + (34.52651153 + 19.93389073 + 34.52651153 i - 19.93389073 i) i}{4}$

Step 9.7

Add $34.52651153$ and $19.93389073$ .

$\frac{19.82196537 + 5.31127961 i + (54.46040227 + 34.52651153 i - 19.93389073 i) i}{4}$

Step 9.8

Subtract $19.93389073 i$ from $34.52651153 i$ .

$\frac{19.82196537 + 5.31127961 i + (54.46040227 + 14.5926208 i) i}{4}$

Step 9.9

Apply the distributive property.

$\frac{19.82196537 + 5.31127961 i + 54.46040227 i + 14.5926208 i i}{4}$

Step 9.10

Multiply $14.5926208 i i$ .

Tap for more steps...

Step 9.10.1

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

$\frac{19.82196537 + 5.31127961 i + 54.46040227 i + 14.5926208 (i^{1} i)}{4}$

Step 9.10.2

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

$\frac{19.82196537 + 5.31127961 i + 54.46040227 i + 14.5926208 (i^{1} i^{1})}{4}$

Step 9.10.3

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

$\frac{19.82196537 + 5.31127961 i + 54.46040227 i + 14.5926208 i^{1 + 1}}{4}$

Step 9.10.4

Add $1$ and $1$ .

$\frac{19.82196537 + 5.31127961 i + 54.46040227 i + 14.5926208 i^{2}}{4}$

Step 9.11

Simplify each term.

Tap for more steps...

Step 9.11.1

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

$\frac{19.82196537 + 5.31127961 i + 54.46040227 i + 14.5926208 \cdot - 1}{4}$

Step 9.11.2

Multiply $14.5926208$ by $- 1$ .

$\frac{19.82196537 + 5.31127961 i + 54.46040227 i - 14.5926208}{4}$

Step 10

Simplify by adding terms.

Tap for more steps...

Step 10.1

Subtract $14.5926208$ from $19.82196537$ .

$\frac{5.22934456 + 5.31127961 i + 54.46040227 i}{4}$

Step 10.2

Add $5.31127961 i$ and $54.46040227 i$ .

$\frac{5.22934456 + 59.77168188 i}{4}$

Step 11

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 12

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 13

Substitute the actual values of $a = 0$ and $b = \frac{1}{4}$ .

$| z | = \sqrt{{(\frac{1}{4})}^{2}}$

Step 14

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

$| z | = \frac{1}{4}$

Step 15

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

$θ = \arctan (\frac{\frac{1}{4}}{0})$

Step 16

Since the argument is undefined and $b$ is positive, the angle of the point on the complex plane is $\frac{π}{2}$ .

$θ = \frac{π}{2}$

Step 17

Substitute the values of $θ = \frac{π}{2}$ and $| z | = \frac{1}{4}$ .

$\frac{1}{4 (\cos (\frac{π}{2}) + i \sin (\frac{π}{2}))}$