Trigonometry Examples

${(4 + 4 i)}^{7}$

Step 1

Use the Binomial Theorem.

$4^{7} + 7 \cdot 4^{6} (4 i) + 21 \cdot 4^{5} {(4 i)}^{2} + 35 \cdot 4^{4} {(4 i)}^{3} + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Raise $4$ to the power of $7$ .

$16384 + 7 \cdot 4^{6} (4 i) + 21 \cdot 4^{5} {(4 i)}^{2} + 35 \cdot 4^{4} {(4 i)}^{3} + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.2

Multiply $4^{6}$ by $4$ by adding the exponents.

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Step 2.1.2.1

Move $4$ .

$16384 + 7 \cdot (4 \cdot 4^{6}) i + 21 \cdot 4^{5} {(4 i)}^{2} + 35 \cdot 4^{4} {(4 i)}^{3} + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.2.2

Multiply $4$ by $4^{6}$ .

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Step 2.1.2.2.1

Raise $4$ to the power of $1$ .

$16384 + 7 \cdot (4^{1} \cdot 4^{6}) i + 21 \cdot 4^{5} {(4 i)}^{2} + 35 \cdot 4^{4} {(4 i)}^{3} + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.2.2.2

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

$16384 + 7 \cdot 4^{1 + 6} i + 21 \cdot 4^{5} {(4 i)}^{2} + 35 \cdot 4^{4} {(4 i)}^{3} + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.2.3

Add $1$ and $6$ .

$16384 + 7 \cdot 4^{7} i + 21 \cdot 4^{5} {(4 i)}^{2} + 35 \cdot 4^{4} {(4 i)}^{3} + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.3

Raise $4$ to the power of $7$ .

$16384 + 7 \cdot 16384 i + 21 \cdot 4^{5} {(4 i)}^{2} + 35 \cdot 4^{4} {(4 i)}^{3} + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.4

Multiply $7$ by $16384$ .

$16384 + 114688 i + 21 \cdot 4^{5} {(4 i)}^{2} + 35 \cdot 4^{4} {(4 i)}^{3} + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.5

Raise $4$ to the power of $5$ .

$16384 + 114688 i + 21 \cdot 1024 {(4 i)}^{2} + 35 \cdot 4^{4} {(4 i)}^{3} + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.6

Multiply $21$ by $1024$ .

$16384 + 114688 i + 21504 {(4 i)}^{2} + 35 \cdot 4^{4} {(4 i)}^{3} + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.7

Apply the product rule to $4 i$ .

$16384 + 114688 i + 21504 (4^{2} i^{2}) + 35 \cdot 4^{4} {(4 i)}^{3} + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.8

Raise $4$ to the power of $2$ .

$16384 + 114688 i + 21504 (16 i^{2}) + 35 \cdot 4^{4} {(4 i)}^{3} + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.9

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

$16384 + 114688 i + 21504 (16 \cdot - 1) + 35 \cdot 4^{4} {(4 i)}^{3} + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.10

Multiply $21504 (16 \cdot - 1)$ .

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Step 2.1.10.1

Multiply $16$ by $- 1$ .

$16384 + 114688 i + 21504 \cdot - 16 + 35 \cdot 4^{4} {(4 i)}^{3} + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.10.2

Multiply $21504$ by $- 16$ .

$16384 + 114688 i - 344064 + 35 \cdot 4^{4} {(4 i)}^{3} + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.11

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

$16384 + 114688 i - 344064 + 35 \cdot 256 {(4 i)}^{3} + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.12

Multiply $35$ by $256$ .

$16384 + 114688 i - 344064 + 8960 {(4 i)}^{3} + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.13

Apply the product rule to $4 i$ .

$16384 + 114688 i - 344064 + 8960 (4^{3} i^{3}) + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.14

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

$16384 + 114688 i - 344064 + 8960 (64 i^{3}) + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.15

Factor out $i^{2}$ .

$16384 + 114688 i - 344064 + 8960 (64 (i^{2} \cdot i)) + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.16

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

$16384 + 114688 i - 344064 + 8960 (64 (- 1 \cdot i)) + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.17

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

$16384 + 114688 i - 344064 + 8960 (64 (- i)) + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.18

Multiply $- 1$ by $64$ .

$16384 + 114688 i - 344064 + 8960 (- 64 i) + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.19

Multiply $- 64$ by $8960$ .

$16384 + 114688 i - 344064 - 573440 i + 35 \cdot 4^{3} {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.20

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

$16384 + 114688 i - 344064 - 573440 i + 35 \cdot 64 {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.21

Multiply $35$ by $64$ .

$16384 + 114688 i - 344064 - 573440 i + 2240 {(4 i)}^{4} + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.22

Apply the product rule to $4 i$ .

$16384 + 114688 i - 344064 - 573440 i + 2240 (4^{4} i^{4}) + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.23

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

$16384 + 114688 i - 344064 - 573440 i + 2240 (256 i^{4}) + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.24

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

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Step 2.1.24.1

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

$16384 + 114688 i - 344064 - 573440 i + 2240 (256 {(i^{2})}^{2}) + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.24.2

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

$16384 + 114688 i - 344064 - 573440 i + 2240 (256 {(- 1)}^{2}) + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.24.3

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

$16384 + 114688 i - 344064 - 573440 i + 2240 (256 \cdot 1) + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.25

Multiply $2240 (256 \cdot 1)$ .

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Step 2.1.25.1

Multiply $256$ by $1$ .

$16384 + 114688 i - 344064 - 573440 i + 2240 \cdot 256 + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.25.2

Multiply $2240$ by $256$ .

$16384 + 114688 i - 344064 - 573440 i + 573440 + 21 \cdot 4^{2} {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.26

Raise $4$ to the power of $2$ .

$16384 + 114688 i - 344064 - 573440 i + 573440 + 21 \cdot 16 {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.27

Multiply $21$ by $16$ .

$16384 + 114688 i - 344064 - 573440 i + 573440 + 336 {(4 i)}^{5} + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.28

Apply the product rule to $4 i$ .

$16384 + 114688 i - 344064 - 573440 i + 573440 + 336 (4^{5} i^{5}) + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.29

Raise $4$ to the power of $5$ .

$16384 + 114688 i - 344064 - 573440 i + 573440 + 336 (1024 i^{5}) + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.30

Factor out $i^{4}$ .

$16384 + 114688 i - 344064 - 573440 i + 573440 + 336 (1024 (i^{4} i)) + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.31

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

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Step 2.1.31.1

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

$16384 + 114688 i - 344064 - 573440 i + 573440 + 336 (1024 ({(i^{2})}^{2} i)) + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.31.2

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

$16384 + 114688 i - 344064 - 573440 i + 573440 + 336 (1024 ({(- 1)}^{2} i)) + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.31.3

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

$16384 + 114688 i - 344064 - 573440 i + 573440 + 336 (1024 (1 i)) + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.32

Multiply $i$ by $1$ .

$16384 + 114688 i - 344064 - 573440 i + 573440 + 336 (1024 i) + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.33

Multiply $1024$ by $336$ .

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i + 7 \cdot 4 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.34

Multiply $7$ by $4$ .

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i + 28 {(4 i)}^{6} + {(4 i)}^{7}$

Step 2.1.35

Apply the product rule to $4 i$ .

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i + 28 (4^{6} i^{6}) + {(4 i)}^{7}$

Step 2.1.36

Raise $4$ to the power of $6$ .

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i + 28 (4096 i^{6}) + {(4 i)}^{7}$

Step 2.1.37

Factor out $i^{4}$ .

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i + 28 (4096 (i^{4} i^{2})) + {(4 i)}^{7}$

Step 2.1.38

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

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Step 2.1.38.1

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

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i + 28 (4096 ({(i^{2})}^{2} i^{2})) + {(4 i)}^{7}$

Step 2.1.38.2

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

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i + 28 (4096 ({(- 1)}^{2} i^{2})) + {(4 i)}^{7}$

Step 2.1.38.3

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

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i + 28 (4096 (1 i^{2})) + {(4 i)}^{7}$

Step 2.1.39

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

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i + 28 (4096 i^{2}) + {(4 i)}^{7}$

Step 2.1.40

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

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i + 28 (4096 \cdot - 1) + {(4 i)}^{7}$

Step 2.1.41

Multiply $28 (4096 \cdot - 1)$ .

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Step 2.1.41.1

Multiply $4096$ by $- 1$ .

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i + 28 \cdot - 4096 + {(4 i)}^{7}$

Step 2.1.41.2

Multiply $28$ by $- 4096$ .

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i - 114688 + {(4 i)}^{7}$

Step 2.1.42

Apply the product rule to $4 i$ .

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i - 114688 + 4^{7} i^{7}$

Step 2.1.43

Raise $4$ to the power of $7$ .

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i - 114688 + 16384 i^{7}$

Step 2.1.44

Rewrite $i^{7}$ as $i^{4} (i^{2} \cdot i)$ .

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Step 2.1.44.1

Factor out $i^{4}$ .

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i - 114688 + 16384 (i^{4} i^{3})$

Step 2.1.44.2

Factor out $i^{2}$ .

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i - 114688 + 16384 (i^{4} (i^{2} \cdot i))$

Step 2.1.45

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

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Step 2.1.45.1

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

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i - 114688 + 16384 ({(i^{2})}^{2} (i^{2} \cdot i))$

Step 2.1.45.2

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

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i - 114688 + 16384 ({(- 1)}^{2} (i^{2} \cdot i))$

Step 2.1.45.3

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

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i - 114688 + 16384 (1 (i^{2} \cdot i))$

Step 2.1.46

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

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i - 114688 + 16384 (i^{2} \cdot i)$

Step 2.1.47

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

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i - 114688 + 16384 (- 1 \cdot i)$

Step 2.1.48

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

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i - 114688 + 16384 (- i)$

Step 2.1.49

Multiply $- 1$ by $16384$ .

$16384 + 114688 i - 344064 - 573440 i + 573440 + 344064 i - 114688 - 16384 i$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $344064$ from $16384$ .

$- 327680 + 114688 i - 573440 i + 573440 + 344064 i - 114688 - 16384 i$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add $- 327680$ and $573440$ .

$245760 + 114688 i - 573440 i + 344064 i - 114688 - 16384 i$

Step 2.2.2.2

Subtract $114688$ from $245760$ .

$131072 + 114688 i - 573440 i + 344064 i - 16384 i$

Step 2.2.3

Subtract $573440 i$ from $114688 i$ .

$131072 - 458752 i + 344064 i - 16384 i$

Step 2.2.4

Add $- 458752 i$ and $344064 i$ .

$131072 - 114688 i - 16384 i$

Step 2.2.5

Subtract $16384 i$ from $- 114688 i$ .

$131072 - 131072 i$

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 $a = 131072$ and $b = - 131072$ .

$| z | = \sqrt{{(- 131072)}^{2} + 131072^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

Raise $- 131072$ to the power of $2$ .

$| z | = \sqrt{17179869184 + 131072^{2}}$

Step 6.2

Raise $131072$ to the power of $2$ .

$| z | = \sqrt{17179869184 + 17179869184}$

Step 6.3

Add $17179869184$ and $17179869184$ .

$| z | = \sqrt{34359738368}$

Step 6.4

Rewrite $34359738368$ as $131072^{2} \cdot 2$ .

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Step 6.4.1

Factor $17179869184$ out of $34359738368$ .

$| z | = \sqrt{17179869184 (2)}$

Step 6.4.2

Rewrite $17179869184$ as $131072^{2}$ .

$| z | = \sqrt{131072^{2} \cdot 2}$

Step 6.5

Pull terms out from under the radical.

$| z | = 131072 \sqrt{2}$

Step 7

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

$θ = \arctan (\frac{- 131072}{131072})$

Step 8

Since inverse tangent of $\frac{- 131072}{131072}$ produces an angle in the fourth quadrant, the value of the angle is $- \frac{π}{4}$ .

$θ = - \frac{π}{4}$

Step 9

Substitute the values of $θ = - \frac{π}{4}$ and $| z | = 131072 \sqrt{2}$ .

$131072 \sqrt{2} (\cos (- \frac{π}{4}) + i \sin (- \frac{π}{4}))$