What is the value of (sin(150)+sin(120))/(cos(210)-cos(300)) ?

The value of (sin(150)+sin(120))/(cos(210)-cos(300)) is -1

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(sin(150)+sin(120))/(cos(210)-cos(300))

Solution

\frac{sin ( 15 0 ^{\circ} ) + sin ( 12 0 ^{\circ} )}{cos ( 21 0 ^{\circ} ) - cos ( 30 0 ^{\circ} )}

Solution

- 1

Solution steps

\frac{sin ( 15 0 ^{\circ} ) + sin ( 12 0 ^{\circ} )}{cos ( 21 0 ^{\circ} ) - cos ( 30 0 ^{\circ} )}

Rewrite using trig identities:

cos (21 0^{\circ}) - cos (30 0^{\circ}) = 2 sin (4 5^{\circ}) sin (25 5^{\circ})

cos (21 0^{\circ}) - cos (30 0^{\circ})

Use the Sum to Product identity:

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

= - 2 sin (\frac{21 0 ^{\circ} + 30 0 ^{\circ}}{2}) sin (\frac{21 0 ^{\circ} - 30 0 ^{\circ}}{2})

Simplify

= - 2 sin (25 5^{\circ}) sin (- 4 5^{\circ})

Use the following property:

sin (- x) = - sin (x)

sin (- 4 5^{\circ}) = - sin (4 5^{\circ})

= - 2 sin (25 5^{\circ}) (- sin (4 5^{\circ}))

Simplify

= 2 sin (4 5^{\circ}) sin (25 5^{\circ})

= \frac{sin ( 15 0 ^{\circ} ) + sin ( 12 0 ^{\circ} )}{2 sin ( 4 5 ^{\circ} ) sin ( 25 5 ^{\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:

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:

sin (4 5^{\circ}) = \frac{2}{2}

sin (4 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}

Rewrite using trig identities:

sin (25 5^{\circ}) = \frac{- 2 - 6}{4}

sin (25 5^{\circ})

Rewrite using trig identities:

sin (13 5^{\circ}) cos (12 0^{\circ}) + cos (13 5^{\circ}) sin (12 0^{\circ})

sin (25 5^{\circ})

Write

sin (25 5^{\circ})

sin (13 5^{\circ} + 12 0^{\circ})

= sin (13 5^{\circ} + 12 0^{\circ})

Use the Angle Sum identity:

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

= sin (13 5^{\circ}) cos (12 0^{\circ}) + cos (13 5^{\circ}) sin (12 0^{\circ})

= sin (13 5^{\circ}) cos (12 0^{\circ}) + cos (13 5^{\circ}) sin (12 0^{\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 (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}

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 (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}

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

Simplify

\frac{2}{2} (- \frac{1}{2}) + (- \frac{2}{2}) \frac{3}{2} : \frac{- 2 - 6}{4}

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

Remove parentheses:

(- a) = - a

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

\frac{2}{2} \cdot \frac{1}{2} = \frac{2}{4}

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

Multiply fractions:

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

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

Multiply:

2 \cdot 1 = 2

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{4}

\frac{2}{2} \cdot \frac{3}{2} = \frac{6}{4}

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

Multiply fractions:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 3}{4}

Simplify

23 : 6

23

Apply radical rule:

a b = a \cdot b

23 = 2 \cdot 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

= \frac{6}{4}

= - \frac{2}{4} - \frac{6}{4}

Apply rule

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

= \frac{- 2 - 6}{4}

= \frac{- 2 - 6}{4}

= \frac{\frac{1}{2} + \frac{3}{2}}{2 \cdot \frac{2}{2} \cdot \frac{- 2 - 6}{4}}

Simplify

\frac{\frac{1}{2} + \frac{3}{2}}{2 \cdot \frac{2}{2} \cdot \frac{- 2 - 6}{4}} : - 1

\frac{\frac{1}{2} + \frac{3}{2}}{2 \cdot \frac{2}{2} \cdot \frac{- 2 - 6}{4}}

Combine the fractions

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

Apply rule

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

= \frac{1 + 3}{2}

= \frac{\frac{1 + 3}{2}}{2 \cdot \frac{2}{2} \cdot \frac{- 2 - 6}{4}}

Apply the fraction rule:

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

= \frac{1 + 3}{2 \cdot 2 \cdot \frac{2}{2} \cdot \frac{- 2 - 6}{4}}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1 + 3}{4 \cdot \frac{2}{2} \cdot \frac{- 2 - 6}{4}}

Multiply

4 \cdot \frac{2}{2} \cdot \frac{- 2 - 6}{4} : \frac{- 2 - 6}{2}

4 \cdot \frac{2}{2} \cdot \frac{- 2 - 6}{4}

Multiply fractions:

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

= \frac{2 ( - 2 - 6 ) \cdot 4}{2 \cdot 4}

Cancel the common factor:

4

= \frac{2 ( - 2 - 6 )}{2}

Apply radical rule:

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

= \frac{2 ^{\frac{1}{2}} ( - 2 - 6 )}{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{- 2 - 6}{2 ^{1 - \frac{1}{2}}}

Subtract the numbers:

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

= \frac{- 2 - 6}{2 ^{\frac{1}{2}}}

Apply radical rule:

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

= \frac{- 2 - 6}{2}

= \frac{1 + 3}{\frac{- 2 - 6}{2}}

Apply the fraction rule:

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

= \frac{( 1 + 3 ) 2}{- 2 - 6}

Rationalize

\frac{2 ( 1 + 3 )}{- 2 - 6} : - 1

\frac{2 ( 1 + 3 )}{- 2 - 6}

Multiply by the conjugate

\frac{- 2 + 6}{- 2 + 6}

= \frac{( 1 + 3 ) 2 ( - 2 + 6 )}{( - 2 - 6 ) ( - 2 + 6 )}

(1 + 3) 2 (- 2 + 6) = 4

(1 + 3) 2 (- 2 + 6)

= 2 (1 + 3) (- 2 + 6)

Expand

(1 + 3) (- 2 + 6) : 22

(1 + 3) (- 2 + 6)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = 1, b = 3, c = - 2, d = 6

= 1 \cdot (- 2) + 1 \cdot 6 + 3 (- 2) + 36

Apply minus-plus rules

+ (- a) = - a

= - 1 \cdot 2 + 1 \cdot 6 - 32 + 36

Simplify

- 1 \cdot 2 + 1 \cdot 6 - 32 + 36 : 22

- 1 \cdot 2 + 1 \cdot 6 - 32 + 36

1 \cdot 2 = 2

1 \cdot 2

Multiply:

1 \cdot 2 = 2

= 2

1 \cdot 6 = 6

1 \cdot 6

Multiply:

1 \cdot 6 = 6

= 6

32 = 6

32

Apply radical rule:

a b = a \cdot b

32 = 3 \cdot 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

36 = 32

36

Factor integer

6 = 3 \cdot 2

= 3 3 \cdot 2

Apply radical rule:

3 \cdot 2 = 32

= 332

Apply radical rule:

a a = a

33 = 3

= 32

= - 2 + 6 - 6 + 32

Add similar elements:

- 2 + 32 = 22

= 22 + 6 - 6

Add similar elements:

6 - 6 = 0

= 22

= 22

= 2 \cdot 22

Expand

2 \cdot 22 : 4

2 \cdot 22

Distribute parentheses

= 2 \cdot 22

= 222

Simplify

222 : 4

222

Apply radical rule:

a a = a

22 = 2

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

= 4

= 4

(- 2 - 6) (- 2 + 6) = - 4

(- 2 - 6) (- 2 + 6)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = - 2, b = - 6, c = - 2, d = 6

= (- 2) (- 2) + (- 2) 6 + (- 6) (- 2) + (- 6) 6

Apply minus-plus rules

(- a) (- b) = ab, + (- a) = - a

= 22 - 26 + 62 - 66

Simplify

22 - 26 + 62 - 66 : - 4

22 - 26 + 62 - 66

Add similar elements:

- 26 + 62 = 0

= 22 - 66

Apply radical rule:

a a = a

22 = 2

= 2 - 66

Apply radical rule:

a a = a

66 = 6

= 2 - 6

Subtract the numbers:

2 - 6 = - 4

= - 4

= - 4

= \frac{4}{- 4}

Apply the fraction rule:

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

= - \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= - 1

= - 1

= - 1

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Frequently Asked Questions (FAQ)

What is the value of (sin(150)+sin(120))/(cos(210)-cos(300)) ?
The value of (sin(150)+sin(120))/(cos(210)-cos(300)) is -1