What is the value of (sin(pi/4))/(1-cos(pi/4)) ?

The value of (sin(pi/4))/(1-cos(pi/4)) is sqrt(2)+1

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(sin(pi/4))/(1-cos(pi/4))

Solution

\frac{sin ( \frac{π}{4} )}{1 - cos ( \frac{π}{4} )}

Solution

2 + 1

Decimal Notation

2.41421 \dots

Solution steps

\frac{sin ( \frac{π}{4} )}{1 - 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{\frac{2}{2}}{1 - \frac{2}{2}}

Simplify

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

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

Apply the fraction rule:

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

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

Join

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

Factor

2 - 2 : 2 (2 - 1)

2 - 2

2 = 22

= 22 - 2

Factor out common term

2

= 2 (2 - 1)

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

Cancel

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

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

Apply radical rule:

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

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= \frac{2 - 1}{2}

= \frac{2 - 1}{2}

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

Multiply

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

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

Multiply fractions:

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

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

Apply radical rule:

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= 2 (2 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 2, c = 1

= 22 - 2 \cdot 1

= 22 - 1 \cdot 2

Simplify

22 - 1 \cdot 2 : 2 - 2

22 - 1 \cdot 2

Apply radical rule:

a a = a

22 = 2

= 2 - 1 \cdot 2

Multiply:

1 \cdot 2 = 2

= 2 - 2

= 2 - 2

= \frac{2}{2 - 2}

Factor

2 - 2 : 2 (2 - 1)

2 - 2

2 = 22

= 22 - 2

Factor out common term

2

= 2 (2 - 1)

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

Cancel the common factor:

2

= \frac{1}{2 - 1}

Rationalize

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

\frac{1}{2 - 1}

Multiply by the conjugate

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

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

1 \cdot (2 + 1) = 2 + 1

(2 - 1) (2 + 1) = 1

(2 - 1) (2 + 1)

Apply Difference of Two Squares Formula:

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

a = 2, b = 1

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

Simplify

(2)^{2} - 1^{2} : 1

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

Apply rule

1^{a} = 1

1^{2} = 1

= (2)^{2} - 1

(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

= 2 - 1

Subtract the numbers:

2 - 1 = 1

= 1

= 1

= \frac{2 + 1}{1}

Apply rule

\frac{a}{1} = a

= 2 + 1

= 2 + 1

= 2 + 1

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Frequently Asked Questions (FAQ)

What is the value of (sin(pi/4))/(1-cos(pi/4)) ?
The value of (sin(pi/4))/(1-cos(pi/4)) is sqrt(2)+1