What is the value of (sin(300)+cos(150))/(tan(120)+cot(240)) ?

The value of (sin(300)+cos(150))/(tan(120)+cot(240)) is 3/2

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(sin(300)+cos(150))/(tan(120)+cot(240))

Solution

\frac{sin ( 30 0 ^{\circ} ) + cos ( 15 0 ^{\circ} )}{tan ( 12 0 ^{\circ} ) + cot ( 24 0 ^{\circ} )}

Solution

\frac{3}{2}

Decimal Notation

1.5

Solution steps

\frac{sin ( 30 0 ^{\circ} ) + cos ( 15 0 ^{\circ} )}{tan ( 12 0 ^{\circ} ) + cot ( 24 0 ^{\circ} )}

Simplify

= \frac{- sin ( 30 0 ^{\circ} ) - cos ( 15 0 ^{\circ} )}{- tan ( 12 0 ^{\circ} ) - cot ( 24 0 ^{\circ} )}

cot (24 0^{\circ}) = cot (6 0^{\circ})

cot (24 0^{\circ})

Rewrite

24 0^{\circ}

18 0^{\circ} + 6 0^{\circ}

= cot (18 0^{\circ} + 6 0^{\circ})

Apply the periodicity of

cot

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

cot (18 0^{\circ} + 6 0^{\circ}) = cot (6 0^{\circ})

= cot (6 0^{\circ})

= \frac{- sin ( 30 0 ^{\circ} ) - cos ( 15 0 ^{\circ} )}{- tan ( 12 0 ^{\circ} ) - cot ( 6 0 ^{\circ} )}

Rewrite using trig identities:

sin (30 0^{\circ}) = 2 sin (15 0^{\circ}) cos (15 0^{\circ})

sin (30 0^{\circ})

Write

sin (30 0^{\circ})

sin (2 \cdot 15 0^{\circ})

= sin (2 \cdot 15 0^{\circ})

Use the Double Angle identity:

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

= 2 sin (15 0^{\circ}) cos (15 0^{\circ})

= \frac{- 2 sin ( 15 0 ^{\circ} ) cos ( 15 0 ^{\circ} ) - cos ( 15 0 ^{\circ} )}{- tan ( 12 0 ^{\circ} ) - cot ( 6 0 ^{\circ} )}

Use the following trivial identity:

sin (15 0^{\circ}) = \frac{1}{2}

sin (15 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}

Use the following trivial identity:

cos (15 0^{\circ}) = - \frac{3}{2}

cos (15 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}

Rewrite using trig identities:

tan (12 0^{\circ}) = - 3

tan (12 0^{\circ})

Rewrite using trig identities:

\frac{sin ( 12 0 ^{\circ} )}{cos ( 12 0 ^{\circ} )}

tan (12 0^{\circ})

Use the basic trigonometric identity:

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

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

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

Use the following trivial identity:

cos (12 0^{\circ}) = - \frac{1}{2}

cos (12 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{1}{2}

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

Simplify

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

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

Apply the fraction rule:

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

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

Divide fractions:

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

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

Refine

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

Cancel the common factor:

2

= - 3

= - 3

Use the following trivial identity:

cot (6 0^{\circ}) = \frac{3}{3}

cot (6 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

= \frac{3}{3}

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

Simplify

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

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

Apply rule

- (- a) = a

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

Join

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

3 - \frac{3}{3}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Add similar elements:

33 - 3 = 23

= \frac{2 3}{3}

Apply radical rule:

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

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= \frac{2}{3}

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

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

Multiply:

1 \cdot 3 = 3

= \frac{3}{2}

= \frac{\frac{3}{2} + \frac{3}{2}}{\frac{2}{3}}

Combine the fractions

\frac{3}{2} + \frac{3}{2} : 3

Apply rule

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

= \frac{3 + 3}{2}

Add similar elements:

3 + 3 = 23

= \frac{2 3}{2}

Divide the numbers:

\frac{2}{2} = 1

= 3

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

Apply the fraction rule:

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

= \frac{3 3}{2}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{2}

= \frac{3}{2}

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Frequently Asked Questions (FAQ)

What is the value of (sin(300)+cos(150))/(tan(120)+cot(240)) ?
The value of (sin(300)+cos(150))/(tan(120)+cot(240)) is 3/2