What is the value of 2cos(pi/8)*sin((7pi)/8) ?

The value of 2cos(pi/8)*sin((7pi)/8) is (sqrt(2))/2

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Popular >

2cos(pi/8)*sin((7pi)/8)

Solution

2 cos (\frac{π}{8}) \cdot sin (\frac{7 π}{8})

Solution

\frac{2}{2}

Decimal Notation

0.70710 \dots

Solution steps

2 cos (\frac{π}{8}) sin (\frac{7 π}{8})

Rewrite using trig identities:

sin (π) + sin (\frac{3 π}{4})

2 cos (\frac{π}{8}) sin (\frac{7 π}{8})

Use the Product to Sum identity:

sin (s) cos (t) = \frac{1}{2} (sin (s + t) + sin (s - t))

= 2 \cdot \frac{1}{2} (sin (\frac{7 π}{8} + \frac{π}{8}) + sin (\frac{7 π}{8} - \frac{π}{8}))

Simplify:

\frac{7 π}{8} + \frac{π}{8} = π

\frac{7 π}{8} + \frac{π}{8}

Apply rule

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

= \frac{7 π + π}{8}

Add similar elements:

7 π + π = 8 π

= \frac{8 π}{8}

Divide the numbers:

\frac{8}{8} = 1

= π

Simplify:

\frac{7 π}{8} - \frac{π}{8} = \frac{3 π}{4}

\frac{7 π}{8} - \frac{π}{8}

Apply rule

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

= \frac{7 π - π}{8}

Add similar elements:

7 π - π = 6 π

= \frac{6 π}{8}

Cancel the common factor:

2

= \frac{3 π}{4}

2 \cdot \frac{1}{2} (sin (π) + sin (\frac{3 π}{4})) = sin (π) + sin (\frac{3 π}{4})

2 \cdot \frac{1}{2} (sin (π) + sin (\frac{3 π}{4}))

Multiply fractions:

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

= \frac{1 \cdot 2 ( sin ( π ) + sin ( \frac{3 π}{4} ) )}{2}

Cancel the common factor:

2

= 1 \cdot (sin (π) + sin (\frac{3 π}{4}))

Multiply:

1 \cdot (sin (π) + sin (\frac{3 π}{4})) = (sin (π) + sin (\frac{3 π}{4}))

= (sin (π) + sin (\frac{3 π}{4}))

Remove parentheses:

(a) = a

= sin (π) + sin (\frac{3 π}{4})

= sin (π) + sin (\frac{3 π}{4})

= sin (π) + sin (\frac{3 π}{4})

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{3 π}{4}) = \frac{2}{2}

sin (\frac{3 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}

= 0 + \frac{2}{2}

Simplify

= \frac{2}{2}

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Frequently Asked Questions (FAQ)

What is the value of 2cos(pi/8)*sin((7pi)/8) ?
The value of 2cos(pi/8)*sin((7pi)/8) is (sqrt(2))/2