What is the value of sin(150)+cos(510)+tan(4110)-tan(210) ?

The value of sin(150)+cos(510)+tan(4110)-tan(210) is -(7sqrt(3)-3)/6

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sin(150)+cos(510)+tan(4110)-tan(210)

Solution

sin (15 0^{\circ}) + cos (51 0^{\circ}) + tan (411 0^{\circ}) - tan (21 0^{\circ})

Solution

- \frac{7 3 - 3}{6}

Decimal Notation

- 1.52072 \dots

Solution steps

sin (15 0^{\circ}) + cos (51 0^{\circ}) + tan (411 0^{\circ}) - tan (21 0^{\circ})

cos (51 0^{\circ}) = cos (15 0^{\circ})

cos (51 0^{\circ})

Rewrite

51 0^{\circ}

36 0^{\circ} + 15 0^{\circ}

= cos (36 0^{\circ} + 15 0^{\circ})

Apply the periodicity of

cos

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

cos (36 0^{\circ} + 15 0^{\circ}) = cos (15 0^{\circ})

= cos (15 0^{\circ})

= sin (15 0^{\circ}) + cos (15 0^{\circ}) + tan (411 0^{\circ}) - tan (21 0^{\circ})

tan (411 0^{\circ}) = tan (15 0^{\circ})

tan (411 0^{\circ})

Rewrite

411 0^{\circ}

18 0^{\circ} \cdot 22 + 15 0^{\circ}

= tan (18 0^{\circ} 22 + 15 0^{\circ})

Apply the periodicity of

tan

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

tan (18 0^{\circ} \cdot 22 + 15 0^{\circ}) = tan (15 0^{\circ})

= tan (15 0^{\circ})

= sin (15 0^{\circ}) + cos (15 0^{\circ}) + tan (15 0^{\circ}) - tan (21 0^{\circ})

tan (21 0^{\circ}) = tan (3 0^{\circ})

tan (21 0^{\circ})

Rewrite

21 0^{\circ}

18 0^{\circ} + 3 0^{\circ}

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

Apply the periodicity of

tan

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

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

= tan (3 0^{\circ})

= sin (15 0^{\circ}) + cos (15 0^{\circ}) + tan (15 0^{\circ}) - tan (3 0^{\circ})

Rewrite using trig identities:

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

tan (15 0^{\circ})

Use the basic trigonometric identity:

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

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

= sin (15 0^{\circ}) + cos (15 0^{\circ}) + \frac{sin ( 15 0 ^{\circ} )}{cos ( 15 0 ^{\circ} )} - tan (3 0^{\circ})

Simplify

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

Use the following trivial identity:

tan (3 0^{\circ}) = \frac{3}{3}

tan (3 0^{\circ})

tan (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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= \frac{3}{3}

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

Simplify

\frac{\frac{1}{2} ( - \frac{3}{2} ) + ( - \frac{3}{2} ) ^{2} + \frac{1}{2} - \frac{3}{3} ( - \frac{3}{2} )}{- \frac{3}{2}} : - \frac{7 3 - 3}{6}

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

Remove parentheses:

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

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

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

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

Group like terms

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

Add similar elements:

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

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

Factor out common term

\frac{3}{2}

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

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

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

Cancel

\frac{3}{3} : \frac{1}{3}

\frac{3}{3}

Cancel

\frac{3}{3} : \frac{1}{3}

\frac{3}{3}

Apply radical rule:

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= \frac{1}{3}

= \frac{1}{3}

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

Least Common Multiplier of

2, 3 : 23

2, 3

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

2

3

= 23

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

23

For

\frac{1}{2} :

multiply the denominator and numerator by

3

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

For

\frac{1}{3} :

multiply the denominator and numerator by

2

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

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

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

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

Apply the fraction rule:

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

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

Apply the fraction rule:

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

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

3

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 3 + 2}{4}

(\frac{3}{2})^{2} = \frac{3}{2 ^{2}}

(\frac{3}{2})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 3 ) ^{2}}{2 ^{2}}

(3)^{2} : 3

Apply radical rule:

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

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 3

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

= - \frac{2 ( \frac{1}{2} + \frac{3}{2 ^{2}} + \frac{2 - 3}{4} )}{3}

2^{2} = 4

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

Combine the fractions

\frac{2 - 3}{4} + \frac{3}{4} : \frac{5 - 3}{4}

Apply rule

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

= \frac{- 3 + 2 + 3}{4}

Add the numbers:

2 + 3 = 5

= \frac{5 - 3}{4}

= - \frac{2 ( \frac{1}{2} + \frac{5 - 3}{4} )}{3}

Join

\frac{5 - 3}{4} + \frac{1}{2} : \frac{7 - 3}{4}

\frac{5 - 3}{4} + \frac{1}{2}

Least Common Multiplier of

4, 2 : 4

4, 2

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

4

2

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

4

For

\frac{1}{2} :

multiply the denominator and numerator by

2

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

= \frac{5 - 3}{4} + \frac{2}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{5 - 3 + 2}{4}

Add the numbers:

5 + 2 = 7

= \frac{7 - 3}{4}

= - \frac{2 \cdot \frac{7 - 3}{4}}{3}

Multiply

\frac{7 - 3}{4} \cdot 2 : \frac{7 - 3}{2}

\frac{7 - 3}{4} \cdot 2

Multiply fractions:

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

= \frac{( 7 - 3 ) \cdot 2}{4}

Cancel the common factor:

2

= \frac{7 - 3}{2}

= - \frac{\frac{7 - 3}{2}}{3}

Apply the fraction rule:

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

= - \frac{7 - 3}{2 3}

Rationalize

- \frac{7 - 3}{2 3} : - \frac{7 3 - 3}{6}

- \frac{7 - 3}{2 3}

Multiply by the conjugate

\frac{3}{3}

= - \frac{( 7 - 3 ) 3}{2 3 3}

(7 - 3) 3 = 73 - 3

(7 - 3) 3

= 3 (7 - 3)

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = 7, c = 3

= 3 \cdot 7 - 33

= 73 - 33

Apply radical rule:

a a = a

33 = 3

= 73 - 3

233 = 6

233

Apply radical rule:

a a = a

33 = 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

= - \frac{7 3 - 3}{6}

= - \frac{7 3 - 3}{6}

= - \frac{7 3 - 3}{6}

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Frequently Asked Questions (FAQ)

What is the value of sin(150)+cos(510)+tan(4110)-tan(210) ?
The value of sin(150)+cos(510)+tan(4110)-tan(210) is -(7sqrt(3)-3)/6