What is the value of (sin(120)cos(210)tan(315)cos(27))/(sin(63)cos(540)) ?

The value of (sin(120)cos(210)tan(315)cos(27))/(sin(63)cos(540)) is -3/4

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(sin(120)cos(210)tan(315)cos(27))/(sin(63)cos(540))

Solution

\frac{sin ( 12 0 ^{\circ} ) cos ( 21 0 ^{\circ} ) tan ( 31 5 ^{\circ} ) cos ( 2 7 ^{\circ} )}{sin ( 6 3 ^{\circ} ) cos ( 54 0 ^{\circ} )}

Solution

- \frac{3}{4}

Decimal Notation

- 0.75

Solution steps

\frac{sin ( 12 0 ^{\circ} ) cos ( 21 0 ^{\circ} ) tan ( 31 5 ^{\circ} ) cos ( 2 7 ^{\circ} )}{sin ( 6 3 ^{\circ} ) cos ( 54 0 ^{\circ} )}

tan (31 5^{\circ}) = tan (13 5^{\circ})

tan (31 5^{\circ})

Rewrite

31 5^{\circ}

18 0^{\circ} + 13 5^{\circ}

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

Apply the periodicity of

tan

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

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

= tan (13 5^{\circ})

= \frac{sin ( 12 0 ^{\circ} ) cos ( 21 0 ^{\circ} ) tan ( 13 5 ^{\circ} ) cos ( 2 7 ^{\circ} )}{sin ( 6 3 ^{\circ} ) cos ( 54 0 ^{\circ} )}

cos (54 0^{\circ}) = cos (18 0^{\circ})

cos (54 0^{\circ})

Rewrite

54 0^{\circ}

36 0^{\circ} + 18 0^{\circ}

= cos (36 0^{\circ} + 18 0^{\circ})

Apply the periodicity of

cos

cos (x + 36 0^{\circ}) = cos (x)

cos (36 0^{\circ} + 18 0^{\circ}) = cos (18 0^{\circ})

= cos (18 0^{\circ})

= \frac{sin ( 12 0 ^{\circ} ) cos ( 21 0 ^{\circ} ) tan ( 13 5 ^{\circ} ) cos ( 2 7 ^{\circ} )}{sin ( 6 3 ^{\circ} ) cos ( 18 0 ^{\circ} )}

Rewrite using trig identities:

sin (6 3^{\circ}) = cos (2 7^{\circ})

sin (6 3^{\circ})

Use the following identity:

sin (x) = cos (9 0^{\circ} - x)

= cos (9 0^{\circ} - 6 3^{\circ})

Simplify

= cos (2 7^{\circ})

= \frac{sin ( 12 0 ^{\circ} ) cos ( 21 0 ^{\circ} ) tan ( 13 5 ^{\circ} ) cos ( 2 7 ^{\circ} )}{cos ( 2 7 ^{\circ} ) cos ( 18 0 ^{\circ} )}

Simplify

= \frac{sin ( 12 0 ^{\circ} ) cos ( 21 0 ^{\circ} ) tan ( 13 5 ^{\circ} )}{cos ( 18 0 ^{\circ} )}

Use the following trivial identity:

sin (12 0^{\circ}) = \frac{3}{2}

sin (12 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}

= \frac{3}{2}

Rewrite using trig identities:

cos (21 0^{\circ}) = - \frac{3}{2}

cos (21 0^{\circ})

Rewrite using trig identities:

cos (18 0^{\circ}) cos (3 0^{\circ}) - sin (18 0^{\circ}) sin (3 0^{\circ})

cos (21 0^{\circ})

Write

cos (21 0^{\circ})

cos (18 0^{\circ} + 3 0^{\circ})

= cos (18 0^{\circ} + 3 0^{\circ})

Use the Angle Sum identity:

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

= cos (18 0^{\circ}) cos (3 0^{\circ}) - sin (18 0^{\circ}) sin (3 0^{\circ})

= cos (18 0^{\circ}) cos (3 0^{\circ}) - sin (18 0^{\circ}) sin (3 0^{\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 (3 0^{\circ}) = \frac{3}{2}

cos (3 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}

= \frac{3}{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 (3 0^{\circ}) = \frac{1}{2}

sin (3 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}

= \frac{1}{2}

= (- 1) \frac{3}{2} - 0 \cdot \frac{1}{2}

Simplify

= - \frac{3}{2}

Rewrite using trig identities:

tan (13 5^{\circ}) = - 1

tan (13 5^{\circ})

Rewrite using trig identities:

\frac{sin ( 13 5 ^{\circ} )}{cos ( 13 5 ^{\circ} )}

tan (13 5^{\circ})

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

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

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

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}

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}

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

Simplify

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

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

Apply the fraction rule:

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

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

Divide fractions:

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

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

Cancel the common factor:

2

= - \frac{2}{2}

Cancel the common factor:

2

= - 1

= - 1

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)

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

Simplify

\frac{\frac{3}{2} ( - \frac{3}{2} ) ( - 1 )}{- 1} : - \frac{3}{4}

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

Remove parentheses:

(- a) = - a, - (- a) = a

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

Multiply:

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

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

Apply the fraction rule:

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

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

Apply the fraction rule:

\frac{a}{1} = a

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

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

Multiply fractions:

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

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

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

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

Multiply the numbers:

2 \cdot 2 = 4

= - \frac{3}{4}

= - \frac{3}{4}

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Frequently Asked Questions (FAQ)

What is the value of (sin(120)cos(210)tan(315)cos(27))/(sin(63)cos(540)) ?
The value of (sin(120)cos(210)tan(315)cos(27))/(sin(63)cos(540)) is -3/4