What is the value of tan(pi/6+(5pi)/4) ?

The value of tan(pi/6+(5pi)/4) is 2+sqrt(3)

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Popular >

tan(pi/6+(5pi)/4)

Solution

tan (\frac{π}{6} + \frac{5 π}{4})

Solution

2 + 3

Decimal Notation

3.73205 \dots

Solution steps

tan (\frac{π}{6} + \frac{5 π}{4})

Rewrite using trig identities:

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

tan (\frac{π}{6} + \frac{5 π}{4})

Use the Angle Sum identity:

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

= \frac{tan ( \frac{π}{6} ) + tan ( \frac{5 π}{4} )}{1 - tan ( \frac{π}{6} ) tan ( \frac{5 π}{4} )}

tan (\frac{5 π}{4}) = tan (\frac{π}{4})

tan (\frac{5 π}{4})

Rewrite

\frac{5 π}{4}

π + \frac{π}{4}

= tan (π + \frac{π}{4})

Apply the periodicity of

tan

tan (x + π) = tan (x)

tan (π + \frac{π}{4}) = tan (\frac{π}{4})

= tan (\frac{π}{4})

= \frac{tan ( \frac{π}{6} ) + tan ( \frac{π}{4} )}{1 - tan ( \frac{π}{6} ) tan ( \frac{5 π}{4} )}

tan (\frac{5 π}{4}) = tan (\frac{π}{4})

tan (\frac{5 π}{4})

Rewrite

\frac{5 π}{4}

π + \frac{π}{4}

= tan (π + \frac{π}{4})

Apply the periodicity of

tan

tan (x + π) = tan (x)

tan (π + \frac{π}{4}) = tan (\frac{π}{4})

= tan (\frac{π}{4})

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

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

Use the following trivial identity:

tan (\frac{π}{4}) = 1

tan (\frac{π}{4})

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}

= 1

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

Simplify

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

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

\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) = 4 + 23

(1 + 3) (3 + 1)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

(1 + 3) (3 + 1) = (1 + 3)^{1 + 1}

= (1 + 3)^{1 + 1}

Add the numbers:

1 + 1 = 2

= (1 + 3)^{2}

Apply Perfect Square Formula:

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

a = 1, b = 3

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

Simplify

1^{2} + 2 \cdot 1 \cdot 3 + (3)^{2} : 4 + 23

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

Apply rule

1^{a} = 1

1^{2} = 1

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

2 \cdot 1 \cdot 3 = 23

2 \cdot 1 \cdot 3

Multiply the numbers:

2 \cdot 1 = 2

= 23

(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

= 1 + 23 + 3

Add the numbers:

1 + 3 = 4

= 4 + 23

= 4 + 23

(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{4 + 2 3}{2}

Factor

4 + 23 : 2 (2 + 3)

4 + 23

Rewrite as

= 2 \cdot 2 + 23

Factor out common term

2

= 2 (2 + 3)

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

Divide the numbers:

\frac{2}{2} = 1

= 2 + 3

= 2 + 3

= 2 + 3

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Frequently Asked Questions (FAQ)

What is the value of tan(pi/6+(5pi)/4) ?
The value of tan(pi/6+(5pi)/4) is 2+sqrt(3)