What is the value of (cos(60)-sin(60)+tan(60))/(sin(60)) ?

The value of (cos(60)-sin(60)+tan(60))/(sin(60)) is (sqrt(3)+3)/3

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(cos(60)-sin(60)+tan(60))/(sin(60))

Solution

\frac{cos ( 6 0 ^{\circ} ) - sin ( 6 0 ^{\circ} ) + tan ( 6 0 ^{\circ} )}{sin ( 6 0 ^{\circ} )}

Solution

\frac{3 + 3}{3}

Decimal Notation

1.57735 \dots

Solution steps

\frac{cos ( 6 0 ^{\circ} ) - sin ( 6 0 ^{\circ} ) + tan ( 6 0 ^{\circ} )}{sin ( 6 0 ^{\circ} )}

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}

Use the following trivial identity:

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 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:

tan (6 0^{\circ}) = 3

tan (6 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}

= 3

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

Simplify

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

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

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

Apply the fraction rule:

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

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

Join

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

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

Convert element to fraction:

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

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

Add similar elements:

- 3 + 23 = 3

= \frac{1 + 3}{2}

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

Multiply

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

= 1 + 3

= \frac{1 + 3}{3}

Rationalize

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

\frac{1 + 3}{3}

Multiply by the conjugate

\frac{3}{3}

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

(1 + 3) 3 = 3 + 3

(1 + 3) 3

= 3 (1 + 3)

Apply the distributive law:

a (b + c) = ab + a c

a = 3, b = 1, c = 3

= 3 \cdot 1 + 33

= 1 \cdot 3 + 33

Simplify

1 \cdot 3 + 33 : 3 + 3

1 \cdot 3 + 33

Multiply:

1 \cdot 3 = 3

= 3 + 33

Apply radical rule:

a a = a

33 = 3

= 3 + 3

= 3 + 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3 + 3}{3}

= \frac{3 + 3}{3}

= \frac{3 + 3}{3}

Popular Examples

arccos(9/41)

arccos (\frac{9}{41})

arccos(1.21)

arccos (1.21)

ln(3)-1/2 ln(5)+arctan(3)-pi/4

ln (3) - \frac{1}{2} ln (5) + arctan (3) - \frac{π}{4}

cos(56.3)

cos (56. 3^{\circ})

arctan(13/15)

arctan (\frac{13}{15})

Frequently Asked Questions (FAQ)

What is the value of (cos(60)-sin(60)+tan(60))/(sin(60)) ?
The value of (cos(60)-sin(60)+tan(60))/(sin(60)) is (sqrt(3)+3)/3