What is the value of (cos(pi)-sec(pi/4)}{cot(pi/6)+sin(\frac{3pi)/2)} ?

The value of (cos(pi)-sec(pi/4)}{cot(pi/6)+sin(\frac{3pi)/2)} is ((-1-sqrt(2))(sqrt(3)+1))/2

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(cos(pi)-sec(pi/4)}{cot(pi/6)+sin(\frac{3pi)/2)}

Solution

\frac{cos ( π ) - sec ( \frac{π}{4} )}{cot ( \frac{π}{6} ) + sin ( \frac{3 π}{2} )}

Solution

\frac{( - 1 - 2 ) ( 3 + 1 )}{2}

Decimal Notation

- 3.29787 \dots

Solution steps

\frac{cos ( π ) - sec ( \frac{π}{4} )}{cot ( \frac{π}{6} ) + sin ( \frac{3 π}{2} )}

Rewrite using trig identities:

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

sin (\frac{3 π}{2})

Write

sin (\frac{3 π}{2})

sin (2 \cdot \frac{3 π}{4})

= sin (2 \cdot \frac{3 π}{4})

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

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

= \frac{cos ( π ) - sec ( \frac{π}{4} )}{cot ( \frac{π}{6} ) + 2 sin ( \frac{3 π}{4} ) cos ( \frac{3 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:

sec (\frac{π}{4}) = 2

sec (\frac{π}{4})

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

= 2

Use the following trivial identity:

cot (\frac{π}{6}) = 3

cot (\frac{π}{6})

cot (x)

periodicity table with

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

= 3

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}

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 - 2}{3 + 2 \cdot \frac{2}{2} ( - \frac{2}{2} )}

Simplify

\frac{- 1 - 2}{3 + 2 \cdot \frac{2}{2} ( - \frac{2}{2} )} : \frac{( - 1 - 2 ) ( 3 + 1 )}{2}

\frac{- 1 - 2}{3 + 2 \cdot \frac{2}{2} ( - \frac{2}{2} )}

Remove parentheses:

(- a) = - a

= \frac{- 1 - 2}{3 - 2 \cdot \frac{2}{2} \cdot \frac{2}{2}}

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

2 \cdot \frac{2}{2} \cdot \frac{2}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= \frac{2 2}{2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

= \frac{- 1 - 2}{3 - 1}

Rationalize

\frac{- 1 - 2}{3 - 1} : \frac{( 1 + 3 ) ( - 1 - 2 )}{2}

\frac{- 1 - 2}{3 - 1}

Multiply by the conjugate

\frac{3 + 1}{3 + 1}

= \frac{( - 1 - 2 ) ( 3 + 1 )}{( 3 - 1 ) ( 3 + 1 )}

(3 - 1) (3 + 1) = 2

(3 - 1) (3 + 1)

Apply Difference of Two Squares Formula:

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

a = 3, b = 1

= (3)^{2} - 1^{2}

Simplify

(3)^{2} - 1^{2} : 2

(3)^{2} - 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= (3)^{2} - 1

(3)^{2} = 3

(3)^{2}

Apply radical rule:

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

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 3^{\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

= 3

= 3 - 1

Subtract the numbers:

3 - 1 = 2

= 2

= 2

= \frac{( - 1 - 2 ) ( 3 + 1 )}{2}

= \frac{( - 1 - 2 ) ( 1 + 3 )}{2}

= \frac{( - 1 - 2 ) ( 3 + 1 )}{2}

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Frequently Asked Questions (FAQ)

What is the value of (cos(pi)-sec(pi/4)}{cot(pi/6)+sin(\frac{3pi)/2)} ?
The value of (cos(pi)-sec(pi/4)}{cot(pi/6)+sin(\frac{3pi)/2)} is ((-1-sqrt(2))(sqrt(3)+1))/2