What is the value of tan((7pi)/6+(3pi)/4) ?

The value of tan((7pi)/6+(3pi)/4) is sqrt(3)-2

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tan((7pi)/6+(3pi)/4)

Solution

tan (\frac{7 π}{6} + \frac{3 π}{4})

Solution

3 - 2

Decimal Notation

- 0.26794 \dots

Solution steps

tan (\frac{7 π}{6} + \frac{3 π}{4})

Rewrite using trig identities:

\frac{tan ( \frac{π}{6} ) + tan ( \frac{3 π}{4} )}{1 - tan ( \frac{π}{6} ) tan ( \frac{3 π}{4} )}

tan (\frac{7 π}{6} + \frac{3 π}{4})

Use the Angle Sum identity:

tan (s + t) = \frac{tan ( s ) + tan ( t )}{1 - tan ( s ) tan ( t )}

= \frac{tan ( \frac{7 π}{6} ) + tan ( \frac{3 π}{4} )}{1 - tan ( \frac{7 π}{6} ) tan ( \frac{3 π}{4} )}

tan (\frac{7 π}{6}) = tan (\frac{π}{6})

tan (\frac{7 π}{6})

Rewrite

\frac{7 π}{6}

π + \frac{π}{6}

= tan (π + \frac{π}{6})

Apply the periodicity of

tan

tan (x + π) = tan (x)

tan (π + \frac{π}{6}) = tan (\frac{π}{6})

= tan (\frac{π}{6})

= \frac{tan ( \frac{π}{6} ) + tan ( \frac{3 π}{4} )}{1 - tan ( \frac{7 π}{6} ) tan ( \frac{3 π}{4} )}

tan (\frac{7 π}{6}) = tan (\frac{π}{6})

tan (\frac{7 π}{6})

Rewrite

\frac{7 π}{6}

π + \frac{π}{6}

= tan (π + \frac{π}{6})

Apply the periodicity of

tan

tan (x + π) = tan (x)

tan (π + \frac{π}{6}) = tan (\frac{π}{6})

= tan (\frac{π}{6})

= \frac{tan ( \frac{π}{6} ) + tan ( \frac{3 π}{4} )}{1 - tan ( \frac{π}{6} ) tan ( \frac{3 π}{4} )}

= \frac{tan ( \frac{π}{6} ) + tan ( \frac{3 π}{4} )}{1 - tan ( \frac{π}{6} ) tan ( \frac{3 π}{4} )}

Use the following trivial identity:

tan (\frac{π}{6}) = \frac{3}{3}

tan (\frac{π}{6})

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= \frac{3}{3}

Rewrite using trig identities:

tan (\frac{3 π}{4}) = - 1

tan (\frac{3 π}{4})

Rewrite using trig identities:

\frac{sin ( \frac{3 π}{4} )}{cos ( \frac{3 π}{4} )}

tan (\frac{3 π}{4})

Use the basic trigonometric identity:

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

= \frac{sin ( \frac{3 π}{4} )}{cos ( \frac{3 π}{4} )}

= \frac{sin ( \frac{3 π}{4} )}{cos ( \frac{3 π}{4} )}

Use the following trivial identity:

sin (\frac{3 π}{4}) = \frac{2}{2}

sin (\frac{3 π}{4})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} 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:

cos (\frac{3 π}{4}) = - \frac{2}{2}

cos (\frac{3 π}{4})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - \frac{2}{2}

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

Simplify

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

\frac{\frac{2}{2}}{- \frac{2}{2}}

Apply the fraction rule:

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

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

Divide fractions:

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

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

Cancel the common factor:

2

= - \frac{2}{2}

Cancel the common factor:

2

= - 1

= - 1

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

Simplify

\frac{\frac{3}{3} - 1}{1 - \frac{3}{3} ( - 1 )} : 3 - 2

\frac{\frac{3}{3} - 1}{1 - \frac{3}{3} ( - 1 )}

Apply rule

- (- a) = a

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

Multiply:

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

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

Join

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

1 + \frac{3}{3}

Convert element to fraction:

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

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

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3 + 3}{3}

Factor

3 + 3 : 3 (3 + 1)

3 + 3

3 = 33

= 33 + 3

Factor out common term

3

= 3 (3 + 1)

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

Cancel

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

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

Apply radical rule:

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= \frac{3 + 1}{3}

= \frac{3 + 1}{3}

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

Join

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

\frac{3}{3} - 1

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3 - 3}{3}

Factor

3 - 3 : 3 (1 - 3)

3 - 3

3 = 33

= 3 - 33

Factor out common term

3

= 3 (1 - 3)

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

Cancel

\frac{3 ( 1 - 3 )}{3} : \frac{1 - 3}{3}

\frac{3 ( 1 - 3 )}{3}

Apply radical rule:

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= \frac{1 - 3}{3}

= \frac{1 - 3}{3}

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

Divide fractions:

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

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

Cancel the common factor:

3

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

Rationalize

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

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

Multiply by the conjugate

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

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

(1 - 3) (3 - 1) = 23 - 4

(1 - 3) (3 - 1)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

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

= 1 \cdot 3 + 1 \cdot (- 1) + (- 3) 3 + (- 3) (- 1)

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

= 1 \cdot 3 - 1 \cdot 1 - 33 + 1 \cdot 3

Simplify

1 \cdot 3 - 1 \cdot 1 - 33 + 1 \cdot 3 : 23 - 4

1 \cdot 3 - 1 \cdot 1 - 33 + 1 \cdot 3

Add similar elements:

1 \cdot 3 + 1 \cdot 3 = 23

= 23 - 1 \cdot 1 - 33

Multiply the numbers:

1 \cdot 1 = 1

= 23 - 1 - 33

Apply radical rule:

a a = a

33 = 3

= 23 - 1 - 3

Subtract the numbers:

- 1 - 3 = - 4

= 23 - 4

= 23 - 4

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

(3 + 1) (3 - 1)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 3, b = 1

= (3)^{2} - 1^{2}

Simplify

(3)^{2} - 1^{2} : 2

(3)^{2} - 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= (3)^{2} - 1

(3)^{2} = 3

(3)^{2}

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

= 3 - 1

Subtract the numbers:

3 - 1 = 2

= 2

= 2

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

Factor

23 - 4 : 2 (3 - 2)

23 - 4

Rewrite as

= 23 - 2 \cdot 2

Factor out common term

2

= 2 (3 - 2)

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

Divide the numbers:

\frac{2}{2} = 1

= 3 - 2

= 3 - 2

= 3 - 2

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Frequently Asked Questions (FAQ)

What is the value of tan((7pi)/6+(3pi)/4) ?
The value of tan((7pi)/6+(3pi)/4) is sqrt(3)-2