What is the value of cos(330)*sin(-210)-tan(-315)*cot(-330) ?

The value of cos(330)*sin(-210)-tan(-315)*cot(-330) is -(3sqrt(3))/4

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cos(330)sin(-210)-tan(-315)cot(-330)

Solution

cos (33 0^{\circ}) \cdot sin (- 21 0^{\circ}) - tan (- 31 5^{\circ}) \cdot cot (- 33 0^{\circ})

Solution

- \frac{3 3}{4}

Decimal Notation

- 1.29903 \dots

Solution steps

cos (33 0^{\circ}) sin (- 21 0^{\circ}) - tan (- 31 5^{\circ}) cot (- 33 0^{\circ})

Use the following property:

cot (- x) = - cot (x)

cot (- 33 0^{\circ}) = - cot (33 0^{\circ})

= cos (33 0^{\circ}) sin (- 21 0^{\circ}) - tan (- 31 5^{\circ}) (- cot (33 0^{\circ}))

Use the following property:

tan (- x) = - tan (x)

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

= cos (33 0^{\circ}) sin (- 21 0^{\circ}) - (- tan (31 5^{\circ})) (- cot (33 0^{\circ}))

Use the following property:

sin (- x) = - sin (x)

sin (- 21 0^{\circ}) = - sin (21 0^{\circ})

= cos (33 0^{\circ}) (- sin (21 0^{\circ})) - (- tan (31 5^{\circ})) (- cot (33 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})

= cos (33 0^{\circ}) (- sin (21 0^{\circ})) - (- tan (13 5^{\circ})) (- cot (33 0^{\circ}))

cot (33 0^{\circ}) = cot (15 0^{\circ})

cot (33 0^{\circ})

Rewrite

33 0^{\circ}

18 0^{\circ} + 15 0^{\circ}

= cot (18 0^{\circ} + 15 0^{\circ})

Apply the periodicity of

cot

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

cot (18 0^{\circ} + 15 0^{\circ}) = cot (15 0^{\circ})

= cot (15 0^{\circ})

= cos (33 0^{\circ}) (- sin (21 0^{\circ})) - (- tan (13 5^{\circ})) (- cot (15 0^{\circ}))

Simplify

= - sin (21 0^{\circ}) cos (33 0^{\circ}) - tan (13 5^{\circ}) cot (15 0^{\circ})

Rewrite using trig identities:

sin (21 0^{\circ}) cos (33 0^{\circ}) = \frac{sin ( 18 0 ^{\circ} ) - sin ( 12 0 ^{\circ} )}{2}

sin (21 0^{\circ}) cos (33 0^{\circ})

Use the Product to Sum identity:

sin (s) cos (t) = \frac{1}{2} (sin (s + t) + sin (s - t))

= \frac{1}{2} (sin (21 0^{\circ} + 33 0^{\circ}) + sin (21 0^{\circ} - 33 0^{\circ}))

Simplify

= \frac{sin ( 54 0 ^{\circ} ) + sin ( - 12 0 ^{\circ} )}{2}

Use the following property:

sin (- x) = - sin (x)

sin (- 12 0^{\circ}) = - sin (12 0^{\circ})

= \frac{sin ( 54 0 ^{\circ} ) - sin ( 12 0 ^{\circ} )}{2}

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

sin (54 0^{\circ})

Rewrite

54 0^{\circ}

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

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

Apply the periodicity of

sin

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

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

= sin (18 0^{\circ})

= \frac{sin ( 18 0 ^{\circ} ) - sin ( 12 0 ^{\circ} )}{2}

= - \frac{sin ( 18 0 ^{\circ} ) - sin ( 12 0 ^{\circ} )}{2} - tan (13 5^{\circ}) cot (15 0^{\circ})

Simplify

= \frac{- sin ( 18 0 ^{\circ} ) + sin ( 12 0 ^{\circ} ) - 2 tan ( 13 5 ^{\circ} ) cot ( 15 0 ^{\circ} )}{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 (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:

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:

cot (15 0^{\circ}) = - 3

cot (15 0^{\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

= - 3

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

Simplify

\frac{- 0 + \frac{3}{2} - 2 ( - 1 ) ( - 3 )}{2} : - \frac{3 3}{4}

\frac{- 0 + \frac{3}{2} - 2 ( - 1 ) ( - 3 )}{2}

Apply rule

- (- a) = a

= \frac{- 0 + \frac{3}{2} - 2 \cdot 1 \cdot 3}{2}

- 0 + \frac{3}{2} - 2 \cdot 1 \cdot 3 = \frac{3}{2} - 23

- 0 + \frac{3}{2} - 2 \cdot 1 \cdot 3

Multiply the numbers:

2 \cdot 1 = 2

= - 0 + \frac{3}{2} - 23

- 0 + \frac{3}{2} - 23 = \frac{3}{2} - 23

= \frac{3}{2} - 23

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

Join

\frac{3}{2} - 23 : - \frac{3 3}{2}

\frac{3}{2} - 23

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

3 - 23 \cdot 2 = - 33

3 - 23 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 3 - 43

Add similar elements:

3 - 43 = - 33

= - 33

= \frac{- 3 3}{2}

Apply the fraction rule:

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

= - \frac{3 3}{2}

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

Apply the fraction rule:

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

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

Apply the fraction rule:

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

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

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

Multiply the numbers:

2 \cdot 2 = 4

= - \frac{3 3}{4}

= - \frac{3 3}{4}

Popular Examples

(200)/(sin(25))

\frac{200}{sin ( 2 5 ^{\circ} )}

tan(1/2+1/3)

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sqrt((100^2+120^2-2*100*120*cos(60)))

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120*sin(60)

120 \cdot sin (6 0^{\circ})

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Frequently Asked Questions (FAQ)

What is the value of cos(330)*sin(-210)-tan(-315)*cot(-330) ?
The value of cos(330)*sin(-210)-tan(-315)*cot(-330) is -(3sqrt(3))/4