What is the value of tan((3pi)/(16)) ?

The value of tan((3pi)/(16)) is sqrt(2\sqrt{2)sqrt(2-\sqrt{2)}-4sqrt(2-\sqrt{2)}+7-4sqrt(2)}

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Popular >

tan((3pi)/(16))

Solution

tan (\frac{3 π}{16})

Solution

22 2 - 2 - 4 2 - 2 + 7 - 42

Decimal Notation

0.66817 \dots

Solution steps

tan (\frac{3 π}{16})

Rewrite using trig identities:

\frac{1 - cos ( \frac{3 π}{8} )}{1 + cos ( \frac{3 π}{8} )}

tan (\frac{3 π}{16})

Write

tan (\frac{3 π}{16})

tan (\frac{\frac{3 π}{8}}{2})

= tan (\frac{\frac{3 π}{8}}{2})

Use the Half Angle identity:

tan (\frac{θ}{2}) = \frac{1 - cos ( θ )}{1 + cos ( θ )}

Rewrite using trig identities:

tan^{2} (θ) = \frac{1 - cos ( 2 θ )}{1 + cos ( 2 θ )}

Use the following identity

tan (θ) = \frac{sin ( θ )}{cos ( θ )}

Square both sides

tan^{2} (θ) = \frac{sin ^{2} ( θ )}{cos ^{2} ( θ )}

Rewrite using trig identities:

sin^{2} (θ) = \frac{1 - cos ( 2 θ )}{2}

Use the Double Angle identity

cos (2 θ) = 1 - 2 sin^{2} (θ)

Switch sides

2 sin^{2} (θ) - 1 = - cos (2 θ)

Add

1

to both sides

2 sin^{2} (θ) = 1 - cos (2 θ)

Divide both sides by

2

sin^{2} (θ) = \frac{1 - cos ( 2 θ )}{2}

Rewrite using trig identities:

cos^{2} (θ) = \frac{1 + cos ( 2 θ )}{2}

Use the Double Angle identity

cos (2 θ) = 2 cos^{2} (θ) - 1

Switch sides

2 cos^{2} (θ) - 1 = cos (2 θ)

Add

1

to both sides

2 sin^{2} (θ) = 1 + cos (2 θ)

Divide both sides by

2

cos^{2} (θ) = \frac{1 + cos ( 2 θ )}{2}

tan^{2} (θ) = \frac{\frac{1 - c o s ( 2 θ )}{2}}{\frac{1 + c o s ( 2 θ )}{2}}

Simplify

tan^{2} (θ) = \frac{1 - cos ( 2 θ )}{1 + cos ( 2 θ )}

Substitute

θ

with

\frac{θ}{2}

tan^{2} (\frac{θ}{2}) = \frac{1 - cos ( 2 \cdot \frac{θ}{2} )}{1 + cos ( 2 \cdot \frac{θ}{2} )}

Simplify

tan^{2} (\frac{θ}{2}) = \frac{1 - cos ( θ )}{1 + cos ( θ )}

Square root both sides

Choose the root sign according to the quadrant of

\frac{θ}{2}

range [0, \frac{π}{2}] [\frac{π}{2}, π] quadrant I II tan positive negative

tan (\frac{θ}{2}) = \frac{1 - cos ( θ )}{1 + cos ( θ )}

= \frac{1 - cos ( \frac{3 π}{8} )}{1 + cos ( \frac{3 π}{8} )}

= \frac{1 - cos ( \frac{3 π}{8} )}{1 + cos ( \frac{3 π}{8} )}

Rewrite using trig identities:

cos (\frac{3 π}{8}) = \frac{2 - 2}{2}

cos (\frac{3 π}{8})

Rewrite using trig identities:

\frac{1 + cos ( \frac{3 π}{4} )}{2}

cos (\frac{3 π}{8})

Write

cos (\frac{3 π}{8})

cos (\frac{\frac{3 π}{4}}{2})

= cos (\frac{\frac{3 π}{4}}{2})

Use the Half Angle identity:

cos (\frac{θ}{2}) = \frac{1 + cos ( θ )}{2}

Use the Double Angle identity

cos (2 θ) = 2 cos^{2} (θ) - 1

Substitute

θ

with

\frac{θ}{2}

cos (θ) = 2 cos^{2} (\frac{θ}{2}) - 1

Switch sides

2 cos^{2} (\frac{θ}{2}) = 1 + cos (θ)

Divide both sides by

2

cos^{2} (\frac{θ}{2}) = \frac{( 1 + cos ( θ ) )}{2}

Square root both sides

Choose the root sign according to the quadrant of

\frac{θ}{2}

range [0, \frac{π}{2}] [\frac{π}{2}, π] [π, \frac{3 π}{2}] [\frac{3 π}{2}, 2 π] quadrant I II III I V sin positive positive negative negative cos positive negative negative positive

cos (\frac{θ}{2}) = \frac{( 1 + cos ( θ ) )}{2}

= \frac{1 + cos ( \frac{3 π}{4} )}{2}

= \frac{1 + cos ( \frac{3 π}{4} )}{2}

Use the following trivial identity:

cos (\frac{3 π}{4}) = - \frac{2}{2}

cos (\frac{3 π}{4})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - \frac{2}{2}

= \frac{1 - \frac{2}{2}}{2}

Simplify

\frac{1 - \frac{2}{2}}{2} : \frac{2 - 2}{2}

\frac{1 - \frac{2}{2}}{2}

\frac{1 - \frac{2}{2}}{2} = \frac{2 - 2}{4}

\frac{1 - \frac{2}{2}}{2}

Join

1 - \frac{2}{2} : \frac{2 - 2}{2}

1 - \frac{2}{2}

Convert element to fraction:

1 = \frac{1 \cdot 2}{2}

= \frac{1 \cdot 2}{2} - \frac{2}{2}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 2 - 2}{2}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 - 2}{2}

= \frac{\frac{2 - 2}{2}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{2 - 2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 - 2}{4}

= \frac{2 - 2}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{2 - 2}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2 - 2}{2}

= \frac{2 - 2}{2}

= \frac{1 - \frac{2 - 2}{2}}{1 + \frac{2 - 2}{2}}

Simplify

\frac{1 - \frac{2 - 2}{2}}{1 + \frac{2 - 2}{2}} : 22 2 - 2 - 4 2 - 2 + 7 - 42

\frac{1 - \frac{2 - 2}{2}}{1 + \frac{2 - 2}{2}}

\frac{1 - \frac{2 - 2}{2}}{1 + \frac{2 - 2}{2}} = \frac{2 - 2 - 2}{2 + 2 - 2}

\frac{1 - \frac{2 - 2}{2}}{1 + \frac{2 - 2}{2}}

Join

1 + \frac{2 - 2}{2} : \frac{2 + 2 - 2}{2}

1 + \frac{2 - 2}{2}

Convert element to fraction:

1 = \frac{1 \cdot 2}{2}

= \frac{1 \cdot 2}{2} + \frac{2 - 2}{2}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 2 + 2 - 2}{2}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 + 2 - 2}{2}

= \frac{1 - \frac{2 - 2}{2}}{\frac{2 + 2 - 2}{2}}

Join

1 - \frac{2 - 2}{2} : \frac{2 - 2 - 2}{2}

1 - \frac{2 - 2}{2}

Convert element to fraction:

1 = \frac{1 \cdot 2}{2}

= \frac{1 \cdot 2}{2} - \frac{2 - 2}{2}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 2 - 2 - 2}{2}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 - 2 - 2}{2}

= \frac{\frac{2 - 2 - 2}{2}}{\frac{2 + 2 - 2}{2}}

Divide fractions:

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}

= \frac{( 2 - 2 - 2 ) \cdot 2}{2 ( 2 + 2 - 2 )}

Cancel the common factor:

2

= \frac{2 - 2 - 2}{2 + 2 - 2}

= \frac{2 - 2 - 2}{2 + 2 - 2}

\frac{2 - 2 - 2}{2 + 2 - 2} = 22 2 - 2 - 4 2 - 2 + 7 - 42

\frac{2 - 2 - 2}{2 + 2 - 2}

Multiply by the conjugate

\frac{2 - 2 - 2}{2 - 2 - 2}

= \frac{( 2 - 2 - 2 ) ( 2 - 2 - 2 )}{( 2 + 2 - 2 ) ( 2 - 2 - 2 )}

(2 - 2 - 2) (2 - 2 - 2) = - 4 2 - 2 + 6 - 2

(2 - 2 - 2) (2 - 2 - 2)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

(2 - 2 - 2) (2 - 2 - 2) = (2 - 2 - 2)^{1 + 1}

= (2 - 2 - 2)^{1 + 1}

Add the numbers:

1 + 1 = 2

= (2 - 2 - 2)^{2}

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 2, b = 2 - 2

= 2^{2} - 2 \cdot 2 2 - 2 + (2 - 2)^{2}

Simplify

2^{2} - 2 \cdot 2 2 - 2 + (2 - 2)^{2} : - 4 2 - 2 + 6 - 2

2^{2} - 2 \cdot 2 2 - 2 + (2 - 2)^{2}

2^{2} = 4

2^{2}

2^{2} = 4

= 4

2 \cdot 2 2 - 2 = 4 2 - 2

2 \cdot 2 2 - 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 2 - 2

(2 - 2)^{2} = 2 - 2

(2 - 2)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= ((2 - 2)^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= (2 - 2)^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2 - 2

= 4 - 4 2 - 2 + 2 - 2

Add the numbers:

4 + 2 = 6

= - 4 2 - 2 + 6 - 2

= - 4 2 - 2 + 6 - 2

(2 + 2 - 2) (2 - 2 - 2) = 2 + 2

(2 + 2 - 2) (2 - 2 - 2)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 2, b = 2 - 2

= 2^{2} - (2 - 2)^{2}

Simplify

2^{2} - (2 - 2)^{2} : 2 + 2

2^{2} - (2 - 2)^{2}

2^{2} = 4

2^{2}

2^{2} = 4

= 4

(2 - 2)^{2} = 2 - 2

(2 - 2)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= ((2 - 2)^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= (2 - 2)^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2 - 2

= 4 - (2 - 2)

- (2 - 2) : - 2 + 2

- (2 - 2)

Distribute parentheses

= - (2) - (- 2)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 2 + 2

= 4 - 2 + 2

Subtract the numbers:

4 - 2 = 2

= 2 + 2

= 2 + 2

= \frac{- 4 2 - 2 + 6 - 2}{2 + 2}

Multiply by the conjugate

\frac{2 - 2}{2 - 2}

= \frac{( - 4 2 - 2 + 6 - 2 ) ( 2 - 2 )}{( 2 + 2 ) ( 2 - 2 )}

(- 4 2 - 2 + 6 - 2) (2 - 2) = 42 2 - 2 - 8 2 - 2 + 14 - 82

(- 4 2 - 2 + 6 - 2) (2 - 2)

Distribute parentheses

= (- 4 2 - 2) \cdot 2 + (- 4 2 - 2) (- 2) + 6 \cdot 2 + 6 (- 2) + (- 2) \cdot 2 + (- 2) (- 2)

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

= - 4 \cdot 2 2 - 2 + 42 2 - 2 + 6 \cdot 2 - 62 - 22 + 22

Simplify

- 4 \cdot 2 2 - 2 + 42 2 - 2 + 6 \cdot 2 - 62 - 22 + 22 : 42 2 - 2 - 8 2 - 2 + 14 - 82

- 4 \cdot 2 2 - 2 + 42 2 - 2 + 6 \cdot 2 - 62 - 22 + 22

Add similar elements:

- 62 - 22 = - 82

= - 4 \cdot 2 2 - 2 + 42 2 - 2 + 6 \cdot 2 - 82 + 22

Multiply the numbers:

4 \cdot 2 = 8

= - 8 2 - 2 + 42 2 - 2 + 6 \cdot 2 - 82 + 22

Multiply the numbers:

6 \cdot 2 = 12

= - 8 2 - 2 + 42 2 - 2 + 12 - 82 + 22

Apply radical rule:

a a = a

22 = 2

= - 8 2 - 2 + 42 2 - 2 + 12 - 82 + 2

Add the numbers:

12 + 2 = 14

= 42 2 - 2 - 8 2 - 2 + 14 - 82

= 42 2 - 2 - 8 2 - 2 + 14 - 82

(2 + 2) (2 - 2) = 2

(2 + 2) (2 - 2)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 2, b = 2

= 2^{2} - (2)^{2}

Simplify

2^{2} - (2)^{2} : 2

2^{2} - (2)^{2}

2^{2} = 4

2^{2}

2^{2} = 4

= 4

(2)^{2} = 2

(2)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2

= 4 - 2

Subtract the numbers:

4 - 2 = 2

= 2

= 2

= \frac{4 2 2 - 2 - 8 2 - 2 + 14 - 8 2}{2}

Factor

42 2 - 2 - 8 2 - 2 + 14 - 82 : 2 (22 - 2 + 2 - 4 2 - 2 + 7 - 42)

42 2 - 2 - 8 2 - 2 + 14 - 82

Rewrite as

= 2 \cdot 22 2 - 2 - 2 \cdot 4 2 - 2 + 2 \cdot 7 - 2 \cdot 42

Factor out common term

2

= 2 (22 2 - 2 - 4 2 - 2 + 7 - 42)

Expand

22 2 - 2 - 4 2 - 2 + 7 - 42 : 22 - 2 + 2 - 4 2 - 2 + 7 - 42

22 2 - 2 - 4 2 - 2 + 7 - 42

22 2 - 2 = 22 - 2 + 2

22 2 - 2

Apply radical rule:

a b = a \cdot b

2 2 - 2 = 2 (2 - 2)

= 2 2 (2 - 2)

Factor

2 - 2 : - (2 - 2)

2 - 2

Factor out common term

- 1

= - (2 - 2)

= 2 - 2 (2 - 2)

Apply radical rule:

assuming

a \geq 0, b \geq 0

- 2 (2 - 2) = 2 - (2 - 2)

= 22 - (2 - 2)

Expand

- (2 - 2) : - 2 + 2

- (2 - 2)

Distribute parentheses

= - (2) - (- 2)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 2 + 2

= 22 2 - 2

= 22 2 - 2 - 4 2 - 2 + 7 - 42

= 2 (22 2 - 2 + 7 - 4 2 - 2 - 42)

= \frac{2 ( 2 2 - 2 + 2 - 4 2 - 2 + 7 - 4 2 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 22 2 - 2 - 4 2 - 2 + 7 - 42

= 22 2 - 2 - 4 2 - 2 + 7 - 42

= 22 2 - 2 - 4 2 - 2 + 7 - 42

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Frequently Asked Questions (FAQ)

What is the value of tan((3pi)/(16)) ?
The value of tan((3pi)/(16)) is sqrt(2\sqrt{2)sqrt(2-\sqrt{2)}-4sqrt(2-\sqrt{2)}+7-4sqrt(2)}