What is the value of (2-sqrt(2sin(pi/4)))/(cos(pi/4)) ?

The value of (2-sqrt(2sin(pi/4)))/(cos(pi/4)) is 2sqrt(2)-2^{3/4}

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(2-sqrt(2sin(pi/4)))/(cos(pi/4))

Solution

\frac{2 - 2 sin ( \frac{π}{4} )}{cos ( \frac{π}{4} )}

Solution

22 - 2^{\frac{3}{4}}

Decimal Notation

1.14663 \dots

Solution steps

\frac{2 - 2 sin ( \frac{π}{4} )}{cos ( \frac{π}{4} )}

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{4})

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

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

Simplify

\frac{2 - 2 \cdot \frac{2}{2}}{\frac{2}{2}} : 22 - 2^{\frac{3}{4}}

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

Apply the fraction rule:

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

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

Apply radical rule:

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

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

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

2 \cdot \frac{2}{2}

Multiply

2 \cdot \frac{2}{2} : 2

2 \cdot \frac{2}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 2

= 2

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

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

Apply the distributive law:

a (b - c) = ab - a c

Apply exponent rule:

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

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

Join

\frac{1}{2} + \frac{1}{4} : \frac{3}{4}

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

Least Common Multiplier of

2, 4 : 4

2, 4

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

2

4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

4

For

\frac{1}{2} :

multiply the denominator and numerator by

2

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

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

Since the denominators are equal, combine the fractions:

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

= \frac{2 + 1}{4}

Add the numbers:

2 + 1 = 3

= \frac{3}{4}

= 2^{\frac{3}{4}}

= 22 - 2^{\frac{3}{4}}

= 22 - 2^{\frac{3}{4}}

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Frequently Asked Questions (FAQ)

What is the value of (2-sqrt(2sin(pi/4)))/(cos(pi/4)) ?
The value of (2-sqrt(2sin(pi/4)))/(cos(pi/4)) is 2sqrt(2)-2^{3/4}