What is the value of (1-cos(135))/(sin(135)) ?

The value of (1-cos(135))/(sin(135)) is sqrt(2)+1

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(1-cos(135))/(sin(135))

Solution

\frac{1 - cos ( 13 5 ^{\circ} )}{sin ( 13 5 ^{\circ} )}

Solution

2 + 1

Decimal Notation

2.41421 \dots

Solution steps

\frac{1 - cos ( 13 5 ^{\circ} )}{sin ( 13 5 ^{\circ} )}

Use the following trivial identity:

cos (13 5^{\circ}) = - \frac{2}{2}

cos (13 5^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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}

Use the following trivial identity:

sin (13 5^{\circ}) = \frac{2}{2}

sin (13 5^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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}

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

Simplify

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

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

Apply rule

- (- a) = a

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

Apply the fraction rule:

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

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

Apply radical rule:

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

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

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

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = 1, c = \frac{2}{2}

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

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

Simplify

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

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

Factor out common term

2

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

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

1 + \frac{2}{2}

Cancel

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

\frac{2}{2}

Cancel

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

\frac{2}{2}

Apply radical rule:

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= \frac{1}{2}

= \frac{1}{2}

= 1 + \frac{1}{2}

Convert element to fraction:

1 = \frac{1 2}{2}

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

Multiply:

1 \cdot 2 = 2

= \frac{2 + 1}{2}

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

Multiply fractions:

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

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

Cancel the common factor:

2

= 2 + 1

= 2 + 1

= 2 + 1

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Frequently Asked Questions (FAQ)

What is the value of (1-cos(135))/(sin(135)) ?
The value of (1-cos(135))/(sin(135)) is sqrt(2)+1