What is the value of 5sin(45)+2tan(30)-4/3 cos(60) ?

The value of 5sin(45)+2tan(30)-4/3 cos(60) is (15sqrt(2)-4+4sqrt(3))/6

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5sin(45)+2tan(30)-4/3 cos(60)

Solution

5 sin (4 5^{\circ}) + 2 tan (3 0^{\circ}) - \frac{4}{3} cos (6 0^{\circ})

Solution

\frac{15 2 - 4 + 4 3}{6}

Decimal Notation

4.02356 \dots

Solution steps

5 sin (4 5^{\circ}) + 2 tan (3 0^{\circ}) - \frac{4}{3} cos (6 0^{\circ})

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}

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}

Use the following trivial identity:

cos (6 0^{\circ}) = \frac{1}{2}

cos (6 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}

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

Simplify

5 \cdot \frac{2}{2} + 2 \cdot \frac{3}{3} - \frac{4}{3} \cdot \frac{1}{2} : \frac{15 2 - 4 + 4 3}{6}

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

Multiply

5 \cdot \frac{2}{2} : \frac{5 2}{2}

5 \cdot \frac{2}{2}

Multiply fractions:

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

= \frac{2 \cdot 5}{2}

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

Multiply

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

2 \cdot \frac{3}{3}

Multiply fractions:

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

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

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

Multiply

\frac{4}{3} \cdot \frac{1}{2} : \frac{2}{3}

\frac{4}{3} \cdot \frac{1}{2}

Cross-cancel common factor:

2

= \frac{2}{3}

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

Combine the fractions

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

Apply rule

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

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

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

Least Common Multiplier of

2, 3 : 6

2, 3

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

2

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

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

6

For

\frac{2 \cdot 5}{2} :

multiply the denominator and numerator by

3

\frac{2 \cdot 5}{2} = \frac{2 \cdot 5 \cdot 3}{2 \cdot 3} = \frac{15 2}{6}

For

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

multiply the denominator and numerator by

2

\frac{3 \cdot 2 - 2}{3} = \frac{( 3 \cdot 2 - 2 ) \cdot 2}{3 \cdot 2} = \frac{( 3 \cdot 2 - 2 ) \cdot 2}{6}

= \frac{15 2}{6} + \frac{( 3 \cdot 2 - 2 ) \cdot 2}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{15 2 + ( 3 \cdot 2 - 2 ) \cdot 2}{6}

Factor

152 + (32 - 2) 2 : 2 (15 + 22 (- 1 + 3))

152 + (3 \cdot 2 - 2) \cdot 2

2 = 22

= 152 + (3 \cdot 2 - 2) 22

Factor out common term

2

= 2 (15 + (- 2 + 23) 2)

Factor

2 (23 - 2) + 15 : 15 + 22 (- 1 + 3)

15 + (- 2 + 23) 2

Factor

- 2 + 23 : 2 (- 1 + 3)

- 2 + 23

Rewrite as

= - 2 \cdot 1 + 23

Factor out common term

2

= 2 (- 1 + 3)

= 15 + 22 (3 - 1)

= 2 (22 (3 - 1) + 15)

= \frac{2 ( 15 + 2 2 ( - 1 + 3 ) )}{6}

Factor

6 : 2 \cdot 3

Factor

6 = 2 \cdot 3

= \frac{2 ( 2 2 ( 3 - 1 ) + 15 )}{2 \cdot 3}

Cancel

\frac{2 ( 15 + 2 2 ( - 1 + 3 ) )}{2 \cdot 3} : \frac{15 + 2 2 ( - 1 + 3 )}{2 \cdot 3}

\frac{2 ( 15 + 2 2 ( - 1 + 3 ) )}{2 \cdot 3}

Apply radical rule:

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

= \frac{2 ^{\frac{1}{2}} ( 2 2 ( 3 - 1 ) + 15 )}{2 \cdot 3}

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{15 + 2 2 ( 3 - 1 )}{3 \cdot 2 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

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

= \frac{15 + 2 2 ( 3 - 1 )}{3 \cdot 2 ^{\frac{1}{2}}}

Apply radical rule:

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

= \frac{15 + 2 2 ( 3 - 1 )}{3 2}

= \frac{15 + 2 2 ( - 1 + 3 )}{2 \cdot 3}

Rationalize

\frac{15 + 2 2 ( 3 - 1 )}{3 2} : \frac{15 2 + 4 3 - 4}{6}

\frac{15 + 2 2 ( 3 - 1 )}{3 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{( 15 + 2 2 ( - 1 + 3 ) ) 2}{2 \cdot 3 2}

(15 + 22 (- 1 + 3)) 2 = 152 - 4 + 43

(15 + 22 (- 1 + 3)) 2

= 2 (15 + 22 (- 1 + 3))

Expand

2 (15 + 22 (- 1 + 3)) : 152 + 4 (- 1 + 3)

2 (15 + 22 (- 1 + 3))

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = 15, c = 22 (- 1 + 3)

= 2 \cdot 15 + 2 \cdot 22 (- 1 + 3)

= 152 + 222 (- 1 + 3)

222 (- 1 + 3) = 4 (- 1 + 3)

222 (- 1 + 3)

Apply radical rule:

a a = a

22 = 2

= 2 \cdot 2 (3 - 1)

Multiply the numbers:

2 \cdot 2 = 4

= 4 (3 - 1)

= 152 + 4 (- 1 + 3)

= 152 + 4 (- 1 + 3)

Expand

4 (- 1 + 3) : - 4 + 43

4 (- 1 + 3)

Apply the distributive law:

a (b + c) = ab + a c

a = 4, b = - 1, c = 3

= 4 (- 1) + 43

Apply minus-plus rules

+ (- a) = - a

= - 4 \cdot 1 + 43

Multiply the numbers:

4 \cdot 1 = 4

= - 4 + 43

= 152 - 4 + 43

2 \cdot 32 = 6

2 \cdot 32

Apply radical rule:

a a = a

22 = 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

= \frac{15 2 - 4 + 4 3}{6}

= \frac{15 2 - 4 + 4 3}{6}

= \frac{15 2 - 4 + 4 3}{6}

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Frequently Asked Questions (FAQ)

What is the value of 5sin(45)+2tan(30)-4/3 cos(60) ?
The value of 5sin(45)+2tan(30)-4/3 cos(60) is (15sqrt(2)-4+4sqrt(3))/6