What is the value of cos(pi/8)+sin(pi/8) ?

The value of cos(pi/8)+sin(pi/8) is sqrt((2+\sqrt{2))/2}

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cos(pi/8)+sin(pi/8)

Solution

cos (\frac{π}{8}) + sin (\frac{π}{8})

Solution

\frac{2 + 2}{2}

Decimal Notation

1.30656 \dots

Solution steps

cos (\frac{π}{8}) + sin (\frac{π}{8})

Rewrite using trig identities:

2 sin (\frac{3 π}{8})

sin (\frac{π}{8}) + cos (\frac{π}{8})

Rewrite as

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

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{1}{2}

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{1}{2}

= 2 (cos (\frac{π}{4}) sin (\frac{π}{8}) + sin (\frac{π}{4}) cos (\frac{π}{8}))

Use the Angle Sum identity:

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

= 2 sin (\frac{π}{8} + \frac{π}{4})

= 2 sin (\frac{π}{8} + \frac{π}{4})

Simplify:

\frac{π}{8} + \frac{π}{4} = \frac{3 π}{8}

\frac{π}{8} + \frac{π}{4}

Least Common Multiplier of

8, 4 : 8

8, 4

Least Common Multiplier (LCM)

Prime factorization of

8 : 2 \cdot 2 \cdot 2

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

8

4

= 2 \cdot 2 \cdot 2

Multiply the numbers:

2 \cdot 2 \cdot 2 = 8

= 8

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

8

For

\frac{π}{4} :

multiply the denominator and numerator by

2

\frac{π}{4} = \frac{π 2}{4 \cdot 2} = \frac{π 2}{8}

= \frac{π}{8} + \frac{π 2}{8}

Since the denominators are equal, combine the fractions:

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

= \frac{π + π 2}{8}

Add similar elements:

π + 2 π = 3 π

= \frac{3 π}{8}

= 2 sin (\frac{3 π}{8})

= 2 sin (\frac{3 π}{8})

Rewrite using trig identities:

sin (\frac{3 π}{8}) = \frac{2 + 2}{2}

sin (\frac{3 π}{8})

Rewrite using trig identities:

\frac{1 - cos ( \frac{3 π}{4} )}{2}

sin (\frac{3 π}{8})

Write

sin (\frac{3 π}{8})

sin (\frac{\frac{3 π}{4}}{2})

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

= \frac{1 - cos ( \frac{3 π}{4} )}{2}

Use the following trivial identity:

cos (\frac{3 π}{4}) = - \frac{2}{2}

cos (\frac{3 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 + 2}{2}

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

Apply rule

- (- a) = a

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

\frac{1 + \frac{2}{2}}{2} = \frac{2 + 2}{4}

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

Join

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

1 + \frac{2}{2}

Convert element to fraction:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 + 2}{2}

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 + 2}{4}

= \frac{2 + 2}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{2 + 2}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2 + 2}{2}

= \frac{2 + 2}{2}

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

Simplify

2 \frac{2 + 2}{2} : \frac{2 + 2}{2}

2 \frac{2 + 2}{2}

Multiply fractions:

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

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

Apply radical rule:

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= \frac{2 + 2}{2}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= \frac{2 + 2}{2}

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

\frac{2 + 2}{2}

Factor

2 + 2 : 2 (2 + 1)

2 + 2

2 = 22

= 22 + 2

Factor out common term

2

= 2 (2 + 1)

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

Cancel

\frac{2 ( 2 + 1 )}{2} : \frac{2 + 1}{2}

\frac{2 ( 2 + 1 )}{2}

Apply radical rule:

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= \frac{2 + 1}{2}

= \frac{2 + 1}{2}

= \frac{2 + 1}{2}

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

\frac{2 + 1}{2}

Multiply by the conjugate

\frac{2}{2}

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

(2 + 1) 2 = 2 + 2

(2 + 1) 2

= 2 (2 + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = 2, c = 1

= 22 + 2 \cdot 1

= 22 + 1 \cdot 2

Simplify

22 + 1 \cdot 2 : 2 + 2

22 + 1 \cdot 2

Apply radical rule:

a a = a

22 = 2

= 2 + 1 \cdot 2

Multiply:

1 \cdot 2 = 2

= 2 + 2

= 2 + 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2 + 2}{2}

= \frac{2 + 2}{2}

= \frac{2 + 2}{2}

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Frequently Asked Questions (FAQ)

What is the value of cos(pi/8)+sin(pi/8) ?
The value of cos(pi/8)+sin(pi/8) is sqrt((2+\sqrt{2))/2}