What is the value of (sec(120)+cot(225)+cos(315))/(sin(270)) ?

The value of (sec(120)+cot(225)+cos(315))/(sin(270)) is (-sqrt(2)+2)/2

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(sec(120)+cot(225)+cos(315))/(sin(270))

Solution

\frac{sec ( 12 0 ^{\circ} ) + cot ( 22 5 ^{\circ} ) + cos ( 31 5 ^{\circ} )}{sin ( 27 0 ^{\circ} )}

Solution

\frac{- 2 + 2}{2}

Decimal Notation

0.29289 \dots

Solution steps

\frac{sec ( 12 0 ^{\circ} ) + cot ( 22 5 ^{\circ} ) + cos ( 31 5 ^{\circ} )}{sin ( 27 0 ^{\circ} )}

Use the following trivial identity:

sec (12 0^{\circ}) = - 2

sec (12 0^{\circ})

sec (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

= - 2

Rewrite using trig identities:

cot (22 5^{\circ}) = 1

cot (22 5^{\circ})

cot (22 5^{\circ}) = cot (4 5^{\circ})

cot (22 5^{\circ})

Rewrite

22 5^{\circ}

18 0^{\circ} + 4 5^{\circ}

= cot (18 0^{\circ} + 4 5^{\circ})

Apply the periodicity of

cot

cot (x + 18 0^{\circ}) = cot (x)

cot (18 0^{\circ} + 4 5^{\circ}) = cot (4 5^{\circ})

= cot (4 5^{\circ})

= cot (4 5^{\circ})

Use the following trivial identity:

cot (4 5^{\circ}) = 1

cot (4 5^{\circ})

cot (x)

periodicity table with

18 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

= 1

= 1

Rewrite using trig identities:

cos (31 5^{\circ}) = \frac{2}{2}

cos (31 5^{\circ})

Rewrite using trig identities:

cos (18 0^{\circ}) cos (13 5^{\circ}) - sin (18 0^{\circ}) sin (13 5^{\circ})

cos (31 5^{\circ})

Write

cos (31 5^{\circ})

cos (18 0^{\circ} + 13 5^{\circ})

= cos (18 0^{\circ} + 13 5^{\circ})

Use the Angle Sum identity:

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

= cos (18 0^{\circ}) cos (13 5^{\circ}) - sin (18 0^{\circ}) sin (13 5^{\circ})

= cos (18 0^{\circ}) cos (13 5^{\circ}) - sin (18 0^{\circ}) sin (13 5^{\circ})

Use the following trivial identity:

cos (18 0^{\circ}) = (- 1)

cos (18 0^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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:

cos (13 5^{\circ}) = - \frac{2}{2}

cos (13 5^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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}

Use the following trivial identity:

sin (18 0^{\circ}) = 0

sin (18 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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 (13 5^{\circ}) = \frac{2}{2}

sin (13 5^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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}

= (- 1) (- \frac{2}{2}) - 0 \cdot \frac{2}{2}

Simplify

= \frac{2}{2}

Rewrite using trig identities:

sin (27 0^{\circ}) = - 1

sin (27 0^{\circ})

Rewrite using trig identities:

sin (18 0^{\circ}) cos (9 0^{\circ}) + cos (18 0^{\circ}) sin (9 0^{\circ})

sin (27 0^{\circ})

Write

sin (27 0^{\circ})

sin (18 0^{\circ} + 9 0^{\circ})

= sin (18 0^{\circ} + 9 0^{\circ})

Use the Angle Sum identity:

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

= sin (18 0^{\circ}) cos (9 0^{\circ}) + cos (18 0^{\circ}) sin (9 0^{\circ})

= sin (18 0^{\circ}) cos (9 0^{\circ}) + cos (18 0^{\circ}) sin (9 0^{\circ})

Use the following trivial identity:

sin (18 0^{\circ}) = 0

sin (18 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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:

cos (9 0^{\circ}) = 0

cos (9 0^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= 0

Use the following trivial identity:

cos (18 0^{\circ}) = (- 1)

cos (18 0^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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:

sin (9 0^{\circ}) = 1

sin (9 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 1

= 0 \cdot 0 + (- 1) \cdot 1

Simplify

= - 1

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

Simplify

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

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

Add/Subtract the numbers:

- 2 + 1 = - 1

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

Apply the fraction rule:

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

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

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

Join

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

- \frac{2}{2} + 1

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

Factor

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

- 2 + 2

2 = 22

= - 2 + 22

Factor out common term

2

= 2 (- 1 + 2)

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

Cancel

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

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

Apply radical rule:

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

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

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

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

Apply the fraction rule:

\frac{a}{1} = a

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

Rationalize

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

\frac{- 1 + 2}{2}

Multiply by the conjugate

\frac{2}{2}

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

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

(- 1 + 2) 2

= 2 (- 1 + 2)

Apply the distributive law:

a (b + c) = ab + a c

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

= 2 (- 1) + 22

Apply minus-plus rules

+ (- a) = - a

= - 1 \cdot 2 + 22

Simplify

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

- 1 \cdot 2 + 22

Multiply:

1 \cdot 2 = 2

= - 2 + 22

Apply radical rule:

a a = a

22 = 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 (sec(120)+cot(225)+cos(315))/(sin(270)) ?
The value of (sec(120)+cot(225)+cos(315))/(sin(270)) is (-sqrt(2)+2)/2