What is the value of cos((pi*(0+1))/8)+0.148*(0)-0.9062 ?

The value of cos((pi*(0+1))/8)+0.148*(0)-0.9062 is (2500sqrt(2+\sqrt{2)}-4531)/(5000)

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cos((pi(0+1))/8)+0.148(0)-0.9062

Solution

cos (\frac{π \cdot ( 0 + 1 )}{8}) + 0.148 \cdot (0) - 0.9062

Solution

\frac{2500 2 + 2 - 4531}{5000}

Decimal Notation

0.01767 \dots

Solution steps

cos (\frac{π ( 0 + 1 )}{8}) + 0.148 (0) - 0.9062

= cos (\frac{π ( 0 + 1 )}{8}) + \frac{37}{250} \cdot 0 - \frac{4531}{5000}

Simplify:

\frac{π ( 0 + 1 )}{8} = \frac{π}{8}

\frac{π ( 0 + 1 )}{8}

π (0 + 1) = π

π (0 + 1)

Add the numbers:

0 + 1 = 1

= 1 π

Multiply:

π 1 = π

= π

= \frac{π}{8}

cos (\frac{π}{8}) + \frac{37}{250} \cdot 0 - \frac{4531}{5000} = \frac{5000 cos ( \frac{π}{8} ) - 4531}{5000}

cos (\frac{π}{8}) + \frac{37}{250} \cdot 0 - \frac{4531}{5000}

Apply rule

0 \cdot a = 0

= cos (\frac{π}{8}) + 0 - \frac{4531}{5000}

cos (\frac{π}{8}) + 0 - \frac{4531}{5000} = cos (\frac{π}{8}) - \frac{4531}{5000}

= cos (\frac{π}{8}) - \frac{4531}{5000}

Convert element to fraction:

cos (\frac{π}{8}) = \frac{cos ( \frac{π}{8} ) 5000}{5000}

= \frac{cos ( \frac{π}{8} ) \cdot 5000}{5000} - \frac{4531}{5000}

Since the denominators are equal, combine the fractions:

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

= \frac{cos ( \frac{π}{8} ) \cdot 5000 - 4531}{5000}

= \frac{5000 cos ( \frac{π}{8} ) - 4531}{5000}

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{5000 \cdot \frac{2 + 2}{2} - 4531}{5000}

Simplify

\frac{5000 \cdot \frac{2 + 2}{2} - 4531}{5000} : \frac{2500 2 + 2 - 4531}{5000}

\frac{5000 \cdot \frac{2 + 2}{2} - 4531}{5000}

5000 \cdot \frac{2 + 2}{2} = 2500 2 + 2

5000 \cdot \frac{2 + 2}{2}

Multiply fractions:

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

= \frac{2 + 2 \cdot 5000}{2}

Divide the numbers:

\frac{5000}{2} = 2500

= 2500 2 + 2

= \frac{2500 2 + 2 - 4531}{5000}

= \frac{2500 2 + 2 - 4531}{5000}

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Frequently Asked Questions (FAQ)

What is the value of cos((pi*(0+1))/8)+0.148*(0)-0.9062 ?
The value of cos((pi*(0+1))/8)+0.148*(0)-0.9062 is (2500sqrt(2+\sqrt{2)}-4531)/(5000)