What is the value of (sin(30)+cos(30))/(tan(30)) ?

The value of (sin(30)+cos(30))/(tan(30)) is (sqrt(3)+3)/2

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Popular >

(sin(30)+cos(30))/(tan(30))

Solution

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

Solution

\frac{3 + 3}{2}

Decimal Notation

2.36602 \dots

Solution steps

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

Use the following trivial identity:

sin (3 0^{\circ}) = \frac{1}{2}

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

cos (3 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}{3}}

Simplify

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

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

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}}{\frac{3}{3}}

Divide fractions:

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

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

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

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

Expand

(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

= \frac{3 + 3}{2}

= \frac{3 + 3}{2}

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Frequently Asked Questions (FAQ)

What is the value of (sin(30)+cos(30))/(tan(30)) ?
The value of (sin(30)+cos(30))/(tan(30)) is (sqrt(3)+3)/2