What is the value of sec(pi/(16)) ?

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Popular >

sec(pi/(16))

Solution

sec (\frac{π}{16})

Solution

4 2 + 2 + 2 + 22 2 + 2 + 2 - 2 2 + 2 + 2 2 + 2 - 2 2 + 2 + 2 2 + 2

Decimal Notation

1.01959 \dots

Solution steps

sec (\frac{π}{16})

Rewrite using trig identities:

\frac{1}{cos ( \frac{π}{16} )}

sec (\frac{π}{16})

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= \frac{1}{cos ( \frac{π}{16} )}

= \frac{1}{cos ( \frac{π}{16} )}

Rewrite using trig identities:

cos (\frac{π}{16}) = \frac{2 + 2 + 2}{2}

cos (\frac{π}{16})

Rewrite using trig identities:

\frac{1 + cos ( \frac{π}{8} )}{2}

cos (\frac{π}{16})

Write

cos (\frac{π}{16})

cos (\frac{\frac{π}{8}}{2})

= cos (\frac{\frac{π}{8}}{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}{8} )}{2}

= \frac{1 + cos ( \frac{π}{8} )}{2}

Rewrite using trig identities:

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

\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}

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

Simplify

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

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

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

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

Join

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

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

Convert element to fraction:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

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

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

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{2 + 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}{2}

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

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

Simplify

\frac{1}{\frac{2 + 2 + 2}{2}} : 4 2 + 2 + 2 + 22 2 + 2 + 2 - 2 2 + 2 + 2 2 + 2 - 2 2 + 2 + 2 2 + 2

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

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

Rationalize

\frac{2}{2 + 2 + 2} : 4 2 + 2 + 2 + 22 2 + 2 + 2 - 2 2 + 2 2 + 2 + 2 - 2 2 + 2 2 + 2 + 2

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

Multiply by the conjugate

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

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

2 + 2 + 2 2 + 2 + 2 = 2 + 2 + 2

2 + 2 + 2 2 + 2 + 2

Apply radical rule:

a a = a

2 + 2 + 2 2 + 2 + 2 = 2 + 2 + 2

= 2 + 2 + 2

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

Multiply by the conjugate

\frac{2 - 2 + 2}{2 - 2 + 2}

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

(2 + 2 + 2) (2 - 2 + 2) = 2 - 2

(2 + 2 + 2) (2 - 2 + 2)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 2, b = 2 + 2

= 2^{2} - (2 + 2)^{2}

Simplify

2^{2} - (2 + 2)^{2} : 2 - 2

2^{2} - (2 + 2)^{2}

2^{2} = 4

2^{2}

2^{2} = 4

= 4

(2 + 2)^{2} = 2 + 2

(2 + 2)^{2}

Apply radical rule:

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

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2 + 2

= 4 - (2 + 2)

- (2 + 2) : - 2 - 2

- (2 + 2)

Distribute parentheses

= - (2) - (2)

Apply minus-plus rules

+ (- a) = - a

= - 2 - 2

= 4 - 2 - 2

Subtract the numbers:

4 - 2 = 2

= 2 - 2

= 2 - 2

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

Cancel

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

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

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

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

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

Multiply by the conjugate

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

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

2 (2 - 2 + 2) 2 + 2 + 2 (2 + 1) = 4 2 + 2 + 2 + 22 2 + 2 + 2 - 2 2 + 2 + 2 2 + 2 - 2 2 + 2 + 2 2 + 2

2 (2 - 2 + 2) 2 + 2 + 2 (2 + 1)

= 2 (2 - 2 + 2) (2 + 1) 2 + 2 + 2

Expand

(2 - 2 + 2) (2 + 1) : 22 + 2 - 2 2 + 2 - 2 + 2

(2 - 2 + 2) (2 + 1)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = 2, b = - 2 + 2, c = 2, d = 1

= 22 + 2 \cdot 1 + (- 2 + 2) 2 + (- 2 + 2) \cdot 1

Apply minus-plus rules

+ (- a) = - a

= 22 + 2 \cdot 1 - 2 2 + 2 - 1 \cdot 2 + 2

Simplify

22 + 2 \cdot 1 - 2 2 + 2 - 1 \cdot 2 + 2 : 22 + 2 - 2 2 + 2 - 2 + 2

22 + 2 \cdot 1 - 2 2 + 2 - 1 \cdot 2 + 2

Multiply the numbers:

2 \cdot 1 = 2

= 22 + 2 - 2 2 + 2 - 1 \cdot 2 + 2

Multiply:

1 \cdot 2 + 2 = 2 + 2

= 22 + 2 - 2 2 + 2 - 2 + 2

= 22 + 2 - 2 2 + 2 - 2 + 2

= 2 2 + 2 + 2 (22 + 2 - 2 2 + 2 - 2 + 2)

Expand

2 2 + 2 + 2 (22 + 2 - 2 2 + 2 - 2 + 2) : 4 2 + 2 + 2 + 22 2 + 2 + 2 - 2 2 + 2 + 2 2 + 2 - 2 2 + 2 + 2 2 + 2

2 2 + 2 + 2 (22 + 2 - 2 2 + 2 - 2 + 2)

Distribute parentheses

= 2 2 + 2 + 2 \cdot 22 + 2 2 + 2 + 2 \cdot 2 + 2 2 + 2 + 2 (- 2 2 + 2) + 2 2 + 2 + 2 (- 2 + 2)

Apply minus-plus rules

+ (- a) = - a

= 222 2 + 2 + 2 + 22 2 + 2 + 2 - 22 2 + 2 + 2 2 + 2 - 2 2 + 2 + 2 2 + 2

Simplify

222 2 + 2 + 2 + 22 2 + 2 + 2 - 22 2 + 2 + 2 2 + 2 - 2 2 + 2 + 2 2 + 2 : 4 2 + 2 + 2 + 22 2 + 2 + 2 - 2 2 + 2 + 2 2 + 2 - 2 2 + 2 + 2 2 + 2

222 2 + 2 + 2 + 22 2 + 2 + 2 - 22 2 + 2 + 2 2 + 2 - 2 2 + 2 + 2 2 + 2

222 2 + 2 + 2 = 4 2 + 2 + 2

222 2 + 2 + 2

Apply radical rule:

a a = a

22 = 2

= 2 \cdot 2 2 + 2 + 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 2 + 2 + 2

22 2 + 2 + 2 2 + 2 = 2 2 + 2 + 2 2 + 2

22 2 + 2 + 2 2 + 2

Apply radical rule:

a a = a

22 = 2

= 2 2 + 2 + 2 2 + 2

= 4 2 + 2 + 2 + 22 2 + 2 + 2 - 2 2 + 2 + 2 2 + 2 - 2 2 + 2 + 2 2 + 2

= 4 2 + 2 + 2 + 22 2 + 2 + 2 - 2 2 + 2 + 2 2 + 2 - 2 2 + 2 + 2 2 + 2

= 4 2 + 2 + 2 + 22 2 + 2 + 2 - 2 2 + 2 + 2 2 + 2 - 2 2 + 2 + 2 2 + 2

(2 - 1) (2 + 1) = 1

(2 - 1) (2 + 1)

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = 2, b = 1

= (2)^{2} - 1^{2}

Simplify

(2)^{2} - 1^{2} : 1

(2)^{2} - 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= (2)^{2} - 1

(2)^{2} = 2

(2)^{2}

Apply radical rule:

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

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2

= 2 - 1

Subtract the numbers:

2 - 1 = 1

= 1

= 1

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

Apply rule

\frac{a}{1} = a

= 4 2 + 2 + 2 + 22 2 + 2 + 2 - 2 2 + 2 + 2 2 + 2 - 2 2 + 2 + 2 2 + 2

= 4 2 + 2 + 2 + 22 2 + 2 + 2 - 2 2 + 2 + 2 2 + 2 - 2 2 + 2 + 2 2 + 2

= 4 2 + 2 + 2 + 22 2 + 2 + 2 - 2 2 + 2 + 2 2 + 2 - 2 2 + 2 + 2 2 + 2

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Frequently Asked Questions (FAQ)

What is the value of sec(pi/(16)) ?
The value of sec(pi/(16)) is