What is the value of sqrt(1+cos^2(pi/8)) ?

The value of sqrt(1+cos^2(pi/8)) is (sqrt(\sqrt{2)+6})/2

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sqrt(1+cos^2(pi/8))

Solution

1 + cos^{2} (\frac{π}{8})

Solution

\frac{2 + 6}{2}

Decimal Notation

1.36145 \dots

Solution steps

1 + cos^{2} (\frac{π}{8})

Rewrite using trig identities:

cos^{2} (\frac{π}{8}) = \frac{1 + cos ( \frac{π}{4} )}{2}

cos^{2} (\frac{π}{8})

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{π}{8} )}{2}

Simplify:

2 \cdot \frac{π}{8} = \frac{π}{4}

2 \cdot \frac{π}{8}

Multiply fractions:

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

= \frac{π 2}{8}

Cancel the common factor:

2

= \frac{π}{4}

= \frac{1 + cos ( \frac{π}{4} )}{2}

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

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

1 + \frac{1 + cos ( \frac{π}{4} )}{2}

Join

1 + \frac{1 + cos ( \frac{π}{4} )}{2} : \frac{cos ( \frac{π}{4} ) + 3}{2}

1 + \frac{1 + cos ( \frac{π}{4} )}{2}

Convert element to fraction:

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

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

1 \cdot 2 + 1 + cos (\frac{π}{4}) = cos (\frac{π}{4}) + 3

1 \cdot 2 + 1 + cos (\frac{π}{4})

Multiply the numbers:

1 \cdot 2 = 2

= 2 + 1 + cos (\frac{π}{4})

Add the numbers:

2 + 1 = 3

= cos (\frac{π}{4}) + 3

= \frac{cos ( \frac{π}{4} ) + 3}{2}

= \frac{cos ( \frac{π}{4} ) + 3}{2}

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

Simplify

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

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

\frac{\frac{2}{2} + 3}{2} = \frac{2 + 6}{4}

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

Join

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

\frac{2}{2} + 3

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

3 \cdot 2 = 6

= \frac{2 + 6}{2}

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

Apply the fraction rule:

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

= \frac{2 + 6}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 + 6}{4}

= \frac{2 + 6}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{2 + 6}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2 + 6}{2}

= \frac{2 + 6}{2}

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Frequently Asked Questions (FAQ)

What is the value of sqrt(1+cos^2(pi/8)) ?
The value of sqrt(1+cos^2(pi/8)) is (sqrt(\sqrt{2)+6})/2