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Popular >

e^{(15pi)/(16)}sin((15pi)/(16))

Solution

e^{\frac{15 π}{16}} sin (\frac{15 π}{16})

Solution

e^{\frac{15 π}{16}} \frac{1 - \frac{2 ^{\frac{1}{2}} + 1}{2 \cdot 2 ^{\frac{1}{2}}}}{2}

Decimal Notation

3.70969 \dots

Solution steps

e^{\frac{15 π}{16}} sin (\frac{15 π}{16})

Rewrite using trig identities:

sin (\frac{15 π}{16}) = sin (\frac{π}{16})

sin (\frac{15 π}{16})

Use the basic trigonometric identity:

sin (x) = sin (π - x)

= sin (π - \frac{15 π}{16})

Simplify:

π - \frac{15 π}{16} = \frac{π}{16}

π - \frac{15 π}{16}

Convert element to fraction:

π = \frac{π 16}{16}

= \frac{π 16}{16} - \frac{15 π}{16}

Since the denominators are equal, combine the fractions:

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

= \frac{π 16 - 15 π}{16}

Add similar elements:

16 π - 15 π = π

= \frac{π}{16}

= sin (\frac{π}{16})

= e^{\frac{15 π}{16}} sin (\frac{π}{16})

Rewrite using trig identities:

sin (\frac{π}{16}) = \frac{1 - \frac{2 ^{\frac{1}{2}} + 1}{2 \cdot 2 ^{\frac{1}{2}}}}{2}

sin (\frac{π}{16})

Rewrite using trig identities:

\frac{1 - cos ( \frac{π}{8} )}{2}

sin (\frac{π}{16})

Write

sin (\frac{π}{16})

sin (\frac{\frac{π}{8}}{2})

= sin (\frac{\frac{π}{8}}{2})

Use the Half Angle identity:

sin (\frac{θ}{2}) = \frac{1 - cos ( θ )}{2}

Use the Double Angle identity

cos (2 θ) = 1 - 2 sin^{2} (θ)

Substitute

θ

with

\frac{θ}{2}

cos (θ) = 1 - 2 sin^{2} (\frac{θ}{2})

Switch sides

2 sin^{2} (\frac{θ}{2}) = 1 - cos (θ)

Divide both sides by

2

sin^{2} (\frac{θ}{2}) = \frac{( 1 - cos ( θ ) )}{2}

Square root both sides

Choose the root sign according to the quadrant of

\frac{θ}{2}

range [0, \frac{π}{2}] [\frac{π}{2}, π] [π, \frac{3 π}{2}] [\frac{3 π}{2}, 2 π] quadrant I II III I V sin positive positive negative negative cos positive negative negative positive

sin (\frac{θ}{2}) = \frac{( 1 - cos ( θ ) )}{2}

= \frac{1 - cos ( \frac{π}{8} )}{2}

= \frac{1 - cos ( \frac{π}{8} )}{2}

Rewrite using trig identities:

cos (\frac{π}{8}) = \frac{2 ^{\frac{1}{2}} + 1}{2 \cdot 2 ^{\frac{1}{2}}}

cos (\frac{π}{8})

Rewrite using trig identities:

\frac{1 + cos ( \frac{π}{4} )}{2}

cos (\frac{π}{8})

Write

cos (\frac{π}{8})

cos (\frac{\frac{π}{4}}{2})

= cos (\frac{\frac{π}{4}}{2})

Use the Half Angle identity:

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

Use the Double Angle identity

cos (2 θ) = 2 cos^{2} (θ) - 1

Substitute

θ

with

\frac{θ}{2}

cos (θ) = 2 cos^{2} (\frac{θ}{2}) - 1

Switch sides

2 cos^{2} (\frac{θ}{2}) = 1 + cos (θ)

Divide both sides by

2

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

Square root both sides

Choose the root sign according to the quadrant of

\frac{θ}{2}

range [0, \frac{π}{2}] [\frac{π}{2}, π] [π, \frac{3 π}{2}] [\frac{3 π}{2}, 2 π] quadrant I II III I V sin positive positive negative negative cos positive negative negative positive

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

= \frac{1 + cos ( \frac{π}{4} )}{2}

= \frac{1 + cos ( \frac{π}{4} )}{2}

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}

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

Simplify

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

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

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

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

Cancel

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

\frac{2}{2}

Cancel

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

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

Join

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

1 + \frac{1}{2 ^{\frac{1}{2}}}

Convert element to fraction:

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

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

Multiply:

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

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

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

Apply the fraction rule:

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

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

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

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

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

= e^{\frac{15 π}{16}} \frac{1 - \frac{2 ^{\frac{1}{2}} + 1}{2 \cdot 2 ^{\frac{1}{2}}}}{2}

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