What is the value of 4cos(pi*1+pi/4) ?

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

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4cos(pi*1+pi/4)

Solution

4 cos (π \cdot 1 + \frac{π}{4})

Solution

- 22

Decimal Notation

- 2.82842 \dots

Solution steps

4 cos (π 1 + \frac{π}{4})

Rewrite using trig identities:

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

cos (π 1 + \frac{π}{4})

Use the Angle Sum identity:

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

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

Simplify

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

= 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}

= 4 ((- 1) \frac{2}{2} - 0 \cdot \frac{2}{2})

Simplify

4 ((- 1) \frac{2}{2} - 0 \cdot \frac{2}{2}) : - 22

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

Remove parentheses:

(- a) = - a

= 4 (- 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}

= 4 (- \frac{2}{2})

Remove parentheses:

(- a) = - a

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

Multiply fractions:

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

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

Divide the numbers:

\frac{4}{2} = 2

= - 22

= - 22

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Frequently Asked Questions (FAQ)

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