What is the value of (tan(60)-tan(30))/(1+tan(60)tan(30)) ?

The value of (tan(60)-tan(30))/(1+tan(60)tan(30)) is (sqrt(3))/3

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

(tan(60)-tan(30))/(1+tan(60)tan(30))

Solution

\frac{tan ( 6 0 ^{\circ} ) - tan ( 3 0 ^{\circ} )}{1 + tan ( 6 0 ^{\circ} ) tan ( 3 0 ^{\circ} )}

Solution

\frac{3}{3}

Decimal Notation

0.57735 \dots

Solution steps

\frac{tan ( 6 0 ^{\circ} ) - tan ( 3 0 ^{\circ} )}{1 + tan ( 6 0 ^{\circ} ) tan ( 3 0 ^{\circ} )}

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

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

Simplify

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

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

3 \frac{3}{3} = 1

3 \frac{3}{3}

Multiply fractions:

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

= \frac{3 3}{3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

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

Join

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

3 - \frac{3}{3}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Add similar elements:

33 - 3 = 23

= \frac{2 3}{3}

Apply radical rule:

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= \frac{2}{3}

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

Add the numbers:

1 + 1 = 2

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

Apply the fraction rule:

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

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

Divide the numbers:

\frac{2}{2} = 1

= \frac{1}{3}

Rationalize

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

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

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

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

= \frac{3}{3}

= \frac{3}{3}

Popular Examples

sin(arctan(-8))

sin (arctan (- 8))

cot((11pi)/4)

cot (\frac{11 π}{4})

cos((3pi)/(10))

cos (\frac{3 π}{10})

csc(-(5pi)/3)

csc (- \frac{5 π}{3})

sin((15pi)/2)

sin (\frac{15 π}{2})

Frequently Asked Questions (FAQ)

What is the value of (tan(60)-tan(30))/(1+tan(60)tan(30)) ?
The value of (tan(60)-tan(30))/(1+tan(60)tan(30)) is (sqrt(3))/3