What is the value of e^{(5pi)/4}cos((5pi)/4) ?

The value of e^{(5pi)/4}cos((5pi)/4) is -(e^{(5pi)/4})/(2^{1/2)}

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e^{(5pi)/4}cos((5pi)/4)

Solution

e^{\frac{5 π}{4}} cos (\frac{5 π}{4})

Solution

- \frac{e ^{\frac{5 π}{4}}}{2 ^{\frac{1}{2}}}

Decimal Notation

- 35.88851 \dots

Solution steps

e^{\frac{5 π}{4}} cos (\frac{5 π}{4})

Rewrite using trig identities:

cos (\frac{5 π}{4}) = - \frac{1}{2 ^{\frac{1}{2}}}

cos (\frac{5 π}{4})

Rewrite using trig identities:

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

cos (\frac{5 π}{4})

Write

cos (\frac{5 π}{4})

cos (π + \frac{π}{4})

= cos (π + \frac{π}{4})

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

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

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

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

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}

= (- 1)

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 (π) = 0

sin (π)

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}

= 0

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}

= (- 1) \frac{2}{2} - 0 \cdot \frac{2}{2}

Simplify

(- 1) \frac{2}{2} - 0 \cdot \frac{2}{2} : - \frac{1}{2 ^{\frac{1}{2}}}

(- 1) \frac{2}{2} - 0 \cdot \frac{2}{2}

Remove parentheses:

(- a) = - a

= - 1 \cdot \frac{2}{2} - 0 \cdot \frac{2}{2}

Add similar elements:

- 1 \cdot \frac{2}{2} - 0 \cdot \frac{2}{2} = - \frac{2}{2}

= - \frac{2}{2}

Apply exponent rule:

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

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

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

= e^{\frac{5 π}{4}} (- \frac{1}{2 ^{\frac{1}{2}}})

Simplify

e^{\frac{5 π}{4}} (- \frac{1}{2 ^{\frac{1}{2}}}) : - \frac{e ^{\frac{5 π}{4}}}{2 ^{\frac{1}{2}}}

e^{\frac{5 π}{4}} (- \frac{1}{2 ^{\frac{1}{2}}})

Remove parentheses:

(- a) = - a

= - e^{\frac{5 π}{4}} \frac{1}{2 ^{\frac{1}{2}}}

Multiply fractions:

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

= - \frac{1 \cdot e ^{\frac{5 π}{4}}}{2 ^{\frac{1}{2}}}

Multiply:

1 \cdot e^{\frac{5 π}{4}} = e^{\frac{5 π}{4}}

= - \frac{e ^{\frac{5 π}{4}}}{2 ^{\frac{1}{2}}}

= - \frac{e ^{\frac{5 π}{4}}}{2 ^{\frac{1}{2}}}

Popular Examples

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cot^{2} (\frac{π}{6})

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Frequently Asked Questions (FAQ)

What is the value of e^{(5pi)/4}cos((5pi)/4) ?
The value of e^{(5pi)/4}cos((5pi)/4) is -(e^{(5pi)/4})/(2^{1/2)}