What is the value of 4sqrt(1+cos^2((7pi)/(32))) ?

The value of 4sqrt(1+cos^2((7pi)/(32))) is 2sqrt(\sqrt{2-\sqrt{2+\sqrt{2)}}+6}

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4sqrt(1+cos^2((7pi)/(32)))

Solution

4 1 + cos^{2} (\frac{7 π}{32})

Solution

2 2 - 2 + 2 + 6

Decimal Notation

5.05576 \dots

Solution steps

4 1 + cos^{2} (\frac{7 π}{32})

Rewrite using trig identities:

cos^{2} (\frac{7 π}{32}) = \frac{1 + cos ( \frac{7 π}{16} )}{2}

cos^{2} (\frac{7 π}{32})

Use the following identity:

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}

= \frac{1 + cos ( 2 \cdot \frac{7 π}{32} )}{2}

Simplify:

2 \cdot \frac{7 π}{32} = \frac{7 π}{16}

2 \cdot \frac{7 π}{32}

Multiply fractions:

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

= \frac{7 π 2}{32}

Multiply the numbers:

7 \cdot 2 = 14

= \frac{14 π}{32}

Cancel the common factor:

2

= \frac{7 π}{16}

= \frac{1 + cos ( \frac{7 π}{16} )}{2}

= 4 1 + \frac{1 + cos ( \frac{7 π}{16} )}{2}

4 1 + \frac{1 + cos ( \frac{7 π}{16} )}{2} = 22 cos (\frac{7 π}{16}) + 3

4 1 + \frac{1 + cos ( \frac{7 π}{16} )}{2}

1 + \frac{1 + cos ( \frac{7 π}{16} )}{2} = \frac{cos ( \frac{7 π}{16} ) + 3}{2}

1 + \frac{1 + cos ( \frac{7 π}{16} )}{2}

Join

1 + \frac{1 + cos ( \frac{7 π}{16} )}{2} : \frac{cos ( \frac{7 π}{16} ) + 3}{2}

1 + \frac{1 + cos ( \frac{7 π}{16} )}{2}

Convert element to fraction:

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

= \frac{1 \cdot 2}{2} + \frac{1 + cos ( \frac{7 π}{16} )}{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 + 1 + cos ( \frac{7 π}{16} )}{2}

1 \cdot 2 + 1 + cos (\frac{7 π}{16}) = cos (\frac{7 π}{16}) + 3

1 \cdot 2 + 1 + cos (\frac{7 π}{16})

Multiply the numbers:

1 \cdot 2 = 2

= 2 + 1 + cos (\frac{7 π}{16})

Add the numbers:

2 + 1 = 3

= cos (\frac{7 π}{16}) + 3

= \frac{cos ( \frac{7 π}{16} ) + 3}{2}

= \frac{cos ( \frac{7 π}{16} ) + 3}{2}

= 4 \frac{cos ( \frac{7 π}{16} ) + 3}{2}

4 = 2 \cdot 2

= 2 \cdot 2 \frac{cos ( \frac{7 π}{16} ) + 3}{2}

Apply exponent rule:

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

2 \frac{cos ( \frac{7 π}{16} ) + 3}{2} = 2^{2} \cdot \frac{cos ( \frac{7 π}{16} ) + 3}{2}

= 2 2^{2} \cdot \frac{cos ( \frac{7 π}{16} ) + 3}{2}

2^{2} \cdot \frac{cos ( \frac{7 π}{16} ) + 3}{2} = 2 cos (\frac{7 π}{16}) + 3

2^{2} \cdot \frac{cos ( \frac{7 π}{16} ) + 3}{2}

Multiply

2^{2} \cdot \frac{cos ( \frac{7 π}{16} ) + 3}{2} : 2 (cos (\frac{7 π}{16}) + 3)

2^{2} \cdot \frac{cos ( \frac{7 π}{16} ) + 3}{2}

Multiply fractions:

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

= \frac{( cos ( \frac{7 π}{16} ) + 3 ) \cdot 2 ^{2}}{2}

Cancel the common factor:

2

= 2 (cos (\frac{7 π}{16}) + 3)

= 2 (cos (\frac{7 π}{16}) + 3)

Apply radical rule:

assuming

a \geq 0, b \geq 0

= 2 cos (\frac{7 π}{16}) + 3

= 22 cos (\frac{7 π}{16}) + 3

= 22 cos (\frac{7 π}{16}) + 3

Rewrite using trig identities:

cos (\frac{7 π}{16}) = \frac{2 - 2 + 2}{2}

cos (\frac{7 π}{16})

Rewrite using trig identities:

\frac{1 + cos ( \frac{7 π}{8} )}{2}

cos (\frac{7 π}{16})

Write

cos (\frac{7 π}{16})

cos (\frac{\frac{7 π}{8}}{2})

= cos (\frac{\frac{7 π}{8}}{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{7 π}{8} )}{2}

= \frac{1 + cos ( \frac{7 π}{8} )}{2}

Rewrite using trig identities:

cos (\frac{7 π}{8}) = - \frac{2 + 2}{2}

cos (\frac{7 π}{8})

Rewrite using trig identities:

- cos (\frac{π}{8})

cos (\frac{7 π}{8})

Use the basic trigonometric identity:

cos (x) = - cos (π - x)

= - cos (π - \frac{7 π}{8})

Simplify:

π - \frac{7 π}{8} = \frac{π}{8}

π - \frac{7 π}{8}

Convert element to fraction:

π = \frac{π 8}{8}

= \frac{π 8}{8} - \frac{7 π}{8}

Since the denominators are equal, combine the fractions:

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

= \frac{π 8 - 7 π}{8}

Add similar elements:

8 π - 7 π = π

= \frac{π}{8}

= - cos (\frac{π}{8})

= - cos (\frac{π}{8})

Rewrite using trig identities:

cos (\frac{π}{8}) = \frac{2 + 2}{2}

cos (\frac{π}{8})

Rewrite using trig identities:

\frac{1 + cos ( \frac{π}{4} )}{2}

cos (\frac{π}{8})

Write

cos (\frac{π}{8})

cos (\frac{\frac{π}{4}}{2})

= cos (\frac{\frac{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{π}{4} )}{2}

= \frac{1 + cos ( \frac{π}{4} )}{2}

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{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{2 + 2}{2}

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

Simplify

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

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

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

\frac{1 - \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}}{2}

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

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

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

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{2 - 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}{2}

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

= 22 \frac{2 - 2 + 2}{2} + 3

Simplify

22 \frac{2 - 2 + 2}{2} + 3 : 2 2 - 2 + 2 + 6

22 \frac{2 - 2 + 2}{2} + 3

Apply radical rule:

a b = a \cdot b

2 \frac{- 2 + 2 + 2}{2} + 3 = 2 \frac{- 2 + 2 + 2}{2} + 3

= 2 2 \frac{- 2 + 2 + 2}{2} + 3

2 \frac{2 - 2 + 2}{2} + 3 = 2 - 2 + 2 + 6

2 \frac{2 - 2 + 2}{2} + 3

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = \frac{2 - 2 + 2}{2}, c = 3

= 2 \cdot \frac{2 - 2 + 2}{2} + 2 \cdot 3

Simplify

2 \cdot \frac{2 - 2 + 2}{2} + 2 \cdot 3 : 2 - 2 + 2 + 6

2 \cdot \frac{2 - 2 + 2}{2} + 2 \cdot 3

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

= 2 - 2 + 2

2 \cdot 3 = 6

2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

= - 2 + 2 + 2 + 6

= 2 - 2 + 2 + 6

= 2 - 2 + 2 + 2 + 6

= 2 2 - 2 + 2 + 6

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Frequently Asked Questions (FAQ)

What is the value of 4sqrt(1+cos^2((7pi)/(32))) ?
The value of 4sqrt(1+cos^2((7pi)/(32))) is 2sqrt(\sqrt{2-\sqrt{2+\sqrt{2)}}+6}