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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

Solution

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

Solution

1 - 2

Decimal Notation

- 0.41421 \dots

Solution steps

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

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}

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}

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

Simplify

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

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

Apply the fraction rule:

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

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

Simplify

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

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

Factor out common term

2

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

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

\frac{2}{2} - 1

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}

= \frac{1}{2} - 1

Convert element to fraction:

1 = \frac{1 2}{2}

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot 2 = 2

= \frac{1 - 2}{2}

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

Multiply fractions:

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

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

Cancel the common factor:

2

= 1 - 2

= 1 - 2

= 1 - 2

Popular Examples

(sec(pi))/2

\frac{sec ( π )}{2}

cos^3(30)

cos^{3} (3 0^{\circ})

sin(arctan((-5)/(12)))

sin (arctan (\frac{- 5}{12}))

2arctan(2)-arcsin(3/5)

2 arctan (2) - arcsin (\frac{3}{5})

4cos(2(0))

4 cos (2 (0))

Frequently Asked Questions (FAQ)

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