What is the value of (27)/(tan(72)) ?

The value of (27)/(tan(72)) is (27(5sqrt(2)-sqrt(10))sqrt(5-\sqrt{5)})/(20)

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(27)/(tan(72))

Solution

\frac{27}{tan ( 7 2 ^{\circ} )}

Solution

\frac{27 ( 5 2 - 10 ) 5 - 5}{20}

Decimal Notation

8.77283 \dots

Solution steps

\frac{27}{tan ( 7 2 ^{\circ} )}

Rewrite using trig identities:

tan (7 2^{\circ}) = \frac{2 tan ( 3 6 ^{\circ} )}{1 - tan ^{2} ( 3 6 ^{\circ} )}

tan (7 2^{\circ})

Write

tan (7 2^{\circ})

tan (2 \cdot 3 6^{\circ})

= tan (2 \cdot 3 6^{\circ})

Use the Double Angle identity:

tan (2 x) = \frac{2 tan ( x )}{1 - tan ^{2} ( x )}

= \frac{2 tan ( 3 6 ^{\circ} )}{1 - tan ^{2} ( 3 6 ^{\circ} )}

= \frac{27}{\frac{2 t a n ( 3 6 ^{\circ} )}{1 - t a n ^{2} ( 3 6 ^{\circ} )}}

Simplify

= \frac{27 ( 1 - tan ^{2} ( 3 6 ^{\circ} ) )}{2 tan ( 3 6 ^{\circ} )}

Rewrite using trig identities:

tan (3 6^{\circ}) = \frac{2 ( 5 - 1 ) 5 - 5}{4}

tan (3 6^{\circ})

Rewrite using trig identities:

\frac{sin ( 3 6 ^{\circ} )}{cos ( 3 6 ^{\circ} )}

tan (3 6^{\circ})

Use the basic trigonometric identity:

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

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

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

Rewrite using trig identities:

sin (3 6^{\circ}) = \frac{2 5 - 5}{4}

sin (3 6^{\circ})

Show that:

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

Use the following product to sum identity:

2 sin (x) cos (y) = sin (x + y) - sin (x - y)

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = sin (5 4^{\circ}) - sin (1 8^{\circ})

Show that:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Use the following identity:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

Divide both sides by

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

Substitute

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

\frac{1}{2} = sin (5 4^{\circ}) - sin (1 8^{\circ})

sin (5 4^{\circ}) = cos (9 0^{\circ} - 5 4^{\circ})

\frac{1}{2} = cos (9 0^{\circ} - 5 4^{\circ}) - sin (1 8^{\circ})

\frac{1}{2} = cos (3 6^{\circ}) - sin (1 8^{\circ})

Show that:

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

Use the factorization rule:

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

a = cos (3 6^{\circ}) + sin (1 8^{\circ})

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = ((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) ((cos (3 6^{\circ}) + sin (1 8^{\circ})) - (cos (3 6^{\circ}) - sin (1 8^{\circ})))

Refine

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 2 (2 cos (3 6^{\circ}) sin (1 8^{\circ}))

Show that:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Use the following identity:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

Divide both sides by

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

Substitute

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 1

Substitute

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (\frac{1}{2})^{2} = 1

Refine

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} = 1

Add

\frac{1}{4}

to both sides

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} + \frac{1}{4} = 1 + \frac{1}{4}

Refine

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} = \frac{5}{4}

Take the square root of both sides

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \pm \frac{5}{4}

cos (3 6^{\circ})

cannot be negative

sin (1 8^{\circ})

cannot be negative

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

Add the following equations

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{2}

((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) = (\frac{5}{2} + \frac{1}{2})

Refine

cos (3 6^{\circ}) = \frac{5 + 1}{4}

Square both sides

(cos (3 6^{\circ}))^{2} = (\frac{5 + 1}{4})^{2}

Use the following identity:

sin^{2} (x) = 1 - cos^{2} (x)

sin^{2} (3 6^{\circ}) = 1 - cos^{2} (3 6^{\circ})

Substitute

cos (3 6^{\circ}) = \frac{5 + 1}{4}

sin^{2} (3 6^{\circ}) = 1 - (\frac{5 + 1}{4})^{2}

Refine

sin^{2} (3 6^{\circ}) = \frac{5 - 5}{8}

Take the square root of both sides

sin (3 6^{\circ}) = \pm \frac{5 - 5}{8}

sin (3 6^{\circ})

cannot be negative

sin (3 6^{\circ}) = \frac{5 - 5}{8}

Refine

sin (3 6^{\circ}) = \frac{\frac{5 - 5}{2}}{2}

= \frac{\frac{5 - 5}{2}}{2}

Simplify

= \frac{2 5 - 5}{4}

Rewrite using trig identities:

cos (3 6^{\circ}) = \frac{5 + 1}{4}

cos (3 6^{\circ})

Show that:

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

Use the following product to sum identity:

2 sin (x) cos (y) = sin (x + y) - sin (x - y)

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = sin (5 4^{\circ}) - sin (1 8^{\circ})

Show that:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Use the following identity:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

Divide both sides by

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

Substitute

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

\frac{1}{2} = sin (5 4^{\circ}) - sin (1 8^{\circ})

sin (5 4^{\circ}) = cos (9 0^{\circ} - 5 4^{\circ})

\frac{1}{2} = cos (9 0^{\circ} - 5 4^{\circ}) - sin (1 8^{\circ})

\frac{1}{2} = cos (3 6^{\circ}) - sin (1 8^{\circ})

Show that:

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

Use the factorization rule:

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

a = cos (3 6^{\circ}) + sin (1 8^{\circ})

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = ((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) ((cos (3 6^{\circ}) + sin (1 8^{\circ})) - (cos (3 6^{\circ}) - sin (1 8^{\circ})))

Refine

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 2 (2 cos (3 6^{\circ}) sin (1 8^{\circ}))

Show that:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Use the following identity:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

Divide both sides by

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

Substitute

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 1

Substitute

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (\frac{1}{2})^{2} = 1

Refine

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} = 1

Add

\frac{1}{4}

to both sides

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} + \frac{1}{4} = 1 + \frac{1}{4}

Refine

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} = \frac{5}{4}

Take the square root of both sides

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \pm \frac{5}{4}

cos (3 6^{\circ})

cannot be negative

sin (1 8^{\circ})

cannot be negative

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

Add the following equations

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{2}

((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) = (\frac{5}{2} + \frac{1}{2})

Refine

cos (3 6^{\circ}) = \frac{5 + 1}{4}

= \frac{5 + 1}{4}

= \frac{\frac{2 5 - 5}{4}}{\frac{5 + 1}{4}}

Simplify

\frac{\frac{2 5 - 5}{4}}{\frac{5 + 1}{4}} : \frac{2 ( 5 - 1 ) 5 - 5}{4}

\frac{\frac{2 5 - 5}{4}}{\frac{5 + 1}{4}}

Divide fractions:

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

= \frac{2 5 - 5 \cdot 4}{4 ( 5 + 1 )}

Cancel the common factor:

4

= \frac{2 5 - 5}{5 + 1}

Rationalize

\frac{2 5 - 5}{5 + 1} : \frac{2 ( 5 - 1 ) 5 - 5}{4}

\frac{2 5 - 5}{5 + 1}

Multiply by the conjugate

\frac{5 - 1}{5 - 1}

= \frac{2 5 - 5 ( 5 - 1 )}{( 5 + 1 ) ( 5 - 1 )}

(5 + 1) (5 - 1) = 4

(5 + 1) (5 - 1)

Apply Difference of Two Squares Formula:

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

a = 5, b = 1

= (5)^{2} - 1^{2}

Simplify

(5)^{2} - 1^{2} : 4

(5)^{2} - 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= (5)^{2} - 1

(5)^{2} = 5

(5)^{2}

Apply radical rule:

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

= (5^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 5^{\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

= 5

= 5 - 1

Subtract the numbers:

5 - 1 = 4

= 4

= 4

= \frac{2 ( 5 - 1 ) 5 - 5}{4}

= \frac{2 ( 5 - 1 ) 5 - 5}{4}

= \frac{2 ( 5 - 1 ) 5 - 5}{4}

= \frac{27 ( 1 - ( \frac{2 ( 5 - 1 ) 5 - 5}{4} ) ^{2} )}{2 \cdot \frac{2 ( 5 - 1 ) 5 - 5}{4}}

Simplify

\frac{27 ( 1 - ( \frac{2 ( 5 - 1 ) 5 - 5}{4} ) ^{2} )}{2 \cdot \frac{2 ( 5 - 1 ) 5 - 5}{4}} : \frac{27 ( 5 2 - 10 ) 5 - 5}{20}

\frac{27 ( 1 - ( \frac{2 ( 5 - 1 ) 5 - 5}{4} ) ^{2} )}{2 \cdot \frac{2 ( 5 - 1 ) 5 - 5}{4}}

Multiply

2 \cdot \frac{2 ( 5 - 1 ) 5 - 5}{4} : \frac{( 5 - 1 ) 5 - 5}{2}

2 \cdot \frac{2 ( 5 - 1 ) 5 - 5}{4}

Multiply fractions:

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

= \frac{2 ( 5 - 1 ) 5 - 5 \cdot 2}{4}

Cancel the common factor:

2

= \frac{2 ( 5 - 1 ) 5 - 5}{2}

Apply radical rule:

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

= \frac{2 ^{\frac{1}{2}} ( 5 - 1 ) 5 - 5}{2}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{( 5 - 1 ) 5 - 5}{2 ^{1 - \frac{1}{2}}}

Subtract the numbers:

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

= \frac{( 5 - 1 ) 5 - 5}{2 ^{\frac{1}{2}}}

Apply radical rule:

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

= \frac{( 5 - 1 ) 5 - 5}{2}

= \frac{27 ( - ( \frac{2 ( 5 - 1 ) 5 - 5}{4} ) ^{2} + 1 )}{\frac{( 5 - 1 ) 5 - 5}{2}}

(\frac{2 ( 5 - 1 ) 5 - 5}{4})^{2} = 5 - 25

(\frac{2 ( 5 - 1 ) 5 - 5}{4})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 2 ( 5 - 1 ) 5 - 5 ) ^{2}}{4 ^{2}}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(2 (5 - 1) 5 - 5)^{2} = (2)^{2} (5 - 5)^{2} (5 - 1)^{2}

= \frac{( 2 ) ^{2} ( 5 - 5 ) ^{2} ( 5 - 1 ) ^{2}}{4 ^{2}}

(2)^{2} : 2

Apply radical rule:

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

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\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

= 2

= \frac{2 ( 5 - 1 ) ^{2} ( 5 - 5 ) ^{2}}{4 ^{2}}

(5 - 5)^{2} : 5 - 5

Apply radical rule:

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

= ((5 - 5)^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= (5 - 5)^{\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

= 5 - 5

= \frac{2 ( 5 - 1 ) ^{2} ( 5 - 5 )}{4 ^{2}}

Factor

4^{2} : 2^{4}

Factor

4 = 2^{2}

= (2^{2})^{2}

Simplify

(2^{2})^{2} : 2^{4}

(2^{2})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= 2^{4}

= 2^{4}

= \frac{2 ( 5 - 1 ) ^{2} ( 5 - 5 )}{2 ^{4}}

Cancel the common factor:

2

= \frac{( 5 - 1 ) ^{2} ( 5 - 5 )}{2 ^{3}}

(5 - 1)^{2} = 6 - 25

(5 - 1)^{2}

Apply Perfect Square Formula:

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

a = 5, b = 1

= (5)^{2} - 25 \cdot 1 + 1^{2}

Simplify

(5)^{2} - 25 \cdot 1 + 1^{2} : 6 - 25

(5)^{2} - 25 \cdot 1 + 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= (5)^{2} - 2 \cdot 1 \cdot 5 + 1

(5)^{2} = 5

(5)^{2}

Apply radical rule:

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

= (5^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 5^{\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

= 5

25 \cdot 1 = 25

25 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 25

= 5 - 25 + 1

Add the numbers:

5 + 1 = 6

= 6 - 25

= 6 - 25

= \frac{( 6 - 2 5 ) ( 5 - 5 )}{2 ^{3}}

Factor

6 - 25 : 2 (3 - 5)

6 - 25

Rewrite as

= 2 \cdot 3 - 25

Factor out common term

2

= 2 (3 - 5)

= \frac{2 ( 3 - 5 ) ( 5 - 5 )}{2 ^{3}}

Cancel the common factor:

2

= \frac{( 3 - 5 ) ( 5 - 5 )}{2 ^{2}}

Expand

(3 - 5) (5 - 5) : 20 - 85

(3 - 5) (5 - 5)

Apply FOIL method:

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

a = 3, b = - 5, c = 5, d = - 5

= 3 \cdot 5 + 3 (- 5) + (- 5) \cdot 5 + (- 5) (- 5)

Apply minus-plus rules

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

= 3 \cdot 5 - 35 - 55 + 55

Simplify

3 \cdot 5 - 35 - 55 + 55 : 20 - 85

3 \cdot 5 - 35 - 55 + 55

Add similar elements:

- 35 - 55 = - 85

= 3 \cdot 5 - 85 + 55

Multiply the numbers:

3 \cdot 5 = 15

= 15 - 85 + 55

Apply radical rule:

a a = a

55 = 5

= 15 - 85 + 5

Add the numbers:

15 + 5 = 20

= 20 - 85

= 20 - 85

= \frac{20 - 8 5}{2 ^{2}}

Factor

20 - 85 : 4 (5 - 25)

20 - 85

Rewrite as

= 4 \cdot 5 - 4 \cdot 25

Factor out common term

4

= 4 (5 - 25)

= \frac{4 ( 5 - 2 5 )}{2 ^{2}}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{2 ^{2} ( 5 - 2 5 )}{2 ^{2}}

Cancel the common factor:

2^{2}

= 5 - 25

= \frac{27 ( - ( 5 - 2 5 ) + 1 )}{\frac{( 5 - 1 ) 5 - 5}{2}}

Apply the fraction rule:

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

= \frac{27 ( 1 - ( 5 - 2 5 ) ) 2}{( 5 - 1 ) 5 - 5}

Expand

1 - (5 - 25) : 25 - 4

1 - (5 - 25)

- (5 - 25) : - 5 + 25

- (5 - 25)

Distribute parentheses

= - (5) - (- 25)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 5 + 25

= 1 - 5 + 25

Subtract the numbers:

1 - 5 = - 4

= 25 - 4

= \frac{27 2 ( 2 5 - 4 )}{( 5 - 1 ) 5 - 5}

Rationalize

\frac{27 2 ( 2 5 - 4 )}{( 5 - 1 ) 5 - 5} : \frac{27 ( 5 2 - 10 ) 5 - 5}{20}

\frac{27 2 ( 2 5 - 4 )}{( 5 - 1 ) 5 - 5}

Multiply by the conjugate

\frac{5 + 1}{5 + 1}

= \frac{27 ( 2 5 - 4 ) 2 ( 5 + 1 )}{( 5 - 1 ) 5 - 5 ( 5 + 1 )}

27 (25 - 4) 2 (5 + 1) = 1622 - 5410

27 (25 - 4) 2 (5 + 1)

= 272 (25 - 4) (5 + 1)

Expand

(25 - 4) (5 + 1) : 6 - 25

(25 - 4) (5 + 1)

Apply FOIL method:

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

a = 25, b = - 4, c = 5, d = 1

= 255 + 25 \cdot 1 + (- 4) 5 + (- 4) \cdot 1

Apply minus-plus rules

+ (- a) = - a

= 255 + 2 \cdot 1 \cdot 5 - 45 - 4 \cdot 1

Simplify

255 + 2 \cdot 1 \cdot 5 - 45 - 4 \cdot 1 : 6 - 25

255 + 2 \cdot 1 \cdot 5 - 45 - 4 \cdot 1

255 = 10

255

Apply radical rule:

a a = a

55 = 5

= 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 10

2 \cdot 1 \cdot 5 = 25

2 \cdot 1 \cdot 5

Multiply the numbers:

2 \cdot 1 = 2

= 25

4 \cdot 1 = 4

4 \cdot 1

Multiply the numbers:

4 \cdot 1 = 4

= 4

= 10 + 25 - 45 - 4

Add similar elements:

25 - 45 = - 25

= 10 - 25 - 4

Subtract the numbers:

10 - 4 = 6

= 6 - 25

= 6 - 25

= 272 (6 - 25)

Expand

272 (6 - 25) : 1622 - 5410

272 (6 - 25)

Apply the distributive law:

a (b - c) = ab - a c

a = 272, b = 6, c = 25

= 272 \cdot 6 - 272 \cdot 25

= 27 \cdot 62 - 27 \cdot 225

Simplify

27 \cdot 62 - 27 \cdot 225 : 1622 - 5410

27 \cdot 62 - 27 \cdot 225

27 \cdot 62 = 1622

27 \cdot 62

Multiply the numbers:

27 \cdot 6 = 162

= 1622

27 \cdot 225 = 5410

27 \cdot 225

Multiply the numbers:

27 \cdot 2 = 54

= 5425

Apply radical rule:

a b = a \cdot b

25 = 2 \cdot 5

= 54 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 5410

= 1622 - 5410

= 1622 - 5410

= 1622 - 5410

(5 - 1) 5 - 5 (5 + 1) = 4 5 - 5

(5 - 1) 5 - 5 (5 + 1)

= (5 - 1) (5 + 1) 5 - 5

Expand

(5 - 1) (5 + 1) : 4

(5 - 1) (5 + 1)

Apply Difference of Two Squares Formula:

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

a = 5, b = 1

= (5)^{2} - 1^{2}

Simplify

(5)^{2} - 1^{2} : 4

(5)^{2} - 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= (5)^{2} - 1

(5)^{2} = 5

(5)^{2}

Apply radical rule:

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

= (5^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 5^{\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

= 5

= 5 - 1

Subtract the numbers:

5 - 1 = 4

= 4

= 4

= 5 - 5 \cdot 4

Expand

5 - 5 \cdot 4 : 4 5 - 5

5 - 5 \cdot 4

Distribute parentheses

= 5 - 5 \cdot 4

= 4 5 - 5

= 4 5 - 5

= \frac{162 2 - 54 10}{4 5 - 5}

Factor

1622 - 5410 : 54 (32 - 10)

1622 - 5410

Rewrite as

= 54 \cdot 32 - 5410

Factor out common term

54

= 54 (32 - 10)

= \frac{54 ( 3 2 - 10 )}{4 5 - 5}

Cancel the common factor:

2

= \frac{27 ( 3 2 - 10 )}{2 5 - 5}

Multiply by the conjugate

\frac{5 - 5}{5 - 5}

= \frac{27 ( 3 2 - 10 ) 5 - 5}{2 5 - 5 5 - 5}

2 5 - 5 5 - 5 = 10 - 25

2 5 - 5 5 - 5

Apply radical rule:

a a = a

5 - 5 5 - 5 = 5 - 5

= 2 (5 - 5)

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 5, c = 5

= 2 \cdot 5 - 25

Multiply the numbers:

2 \cdot 5 = 10

= 10 - 25

= \frac{27 ( 3 2 - 10 ) 5 - 5}{10 - 2 5}

Factor out common term

- 2 : - 2 (5 - 5)

- 25 + 10

Rewrite

10

2 \cdot 5

= - 25 + 2 \cdot 5

Factor out common term

- 2

= - 2 (5 - 5)

= - \frac{27 ( 3 2 - 10 ) 5 - 5}{2 ( 5 - 5 )}

Cancel

- \frac{27 ( 3 2 - 10 ) 5 - 5}{2 ( 5 - 5 )} : \frac{27 ( 3 2 - 10 ) 5 - 5}{2 ( 5 - 5 )}

- \frac{27 ( 3 2 - 10 ) 5 - 5}{2 ( 5 - 5 )}

5 - 5 = - (5 - 5)

= - \frac{27 ( 3 2 - 10 ) 5 - 5}{- 2 ( 5 - 5 )}

Refine

= \frac{27 ( 3 2 - 10 ) 5 - 5}{2 ( 5 - 5 )}

= \frac{27 ( 3 2 - 10 ) 5 - 5}{2 ( 5 - 5 )}

Multiply by the conjugate

\frac{5 + 5}{5 + 5}

= \frac{27 ( 3 2 - 10 ) 5 - 5 ( 5 + 5 )}{2 ( 5 - 5 ) ( 5 + 5 )}

27 (32 - 10) 5 - 5 (5 + 5) = 2702 5 - 5 - 5410 5 - 5

27 (32 - 10) 5 - 5 (5 + 5)

= 27 (32 - 10) (5 + 5) 5 - 5

Expand

(32 - 10) (5 + 5) : 102 - 210

(32 - 10) (5 + 5)

Apply FOIL method:

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

a = 32, b = - 10, c = 5, d = 5

= 32 \cdot 5 + 325 + (- 10) \cdot 5 + (- 10) 5

Apply minus-plus rules

+ (- a) = - a

= 3 \cdot 52 + 325 - 510 - 105

Simplify

3 \cdot 52 + 325 - 510 - 105 : 102 - 210

3 \cdot 52 + 325 - 510 - 105

3 \cdot 52 = 152

3 \cdot 52

Multiply the numbers:

3 \cdot 5 = 15

= 152

325 = 310

325

Apply radical rule:

a b = a \cdot b

25 = 2 \cdot 5

= 3 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 310

105 = 52

105

Factor integer

10 = 5 \cdot 2

= 5 \cdot 2 5

Apply radical rule:

5 \cdot 2 = 52

= 525

Apply radical rule:

a a = a

55 = 5

= 52

= 152 + 310 - 510 - 52

Group like terms

= 152 + 310 - 510 - 52

Add similar elements:

310 - 510 = - 210

= 152 - 210 - 52

Add similar elements:

152 - 52 = 102

= 102 - 210

= 102 - 210

= 27 5 - 5 (102 - 210)

Expand

27 5 - 5 (102 - 210) : 2702 5 - 5 - 5410 5 - 5

27 5 - 5 (102 - 210)

Apply the distributive law:

a (b - c) = ab - a c

a = 27 5 - 5, b = 102, c = 210

= 27 5 - 5 \cdot 102 - 27 5 - 5 \cdot 210

= 27 \cdot 102 5 - 5 - 27 \cdot 210 5 - 5

Simplify

27 \cdot 102 5 - 5 - 27 \cdot 210 5 - 5 : 2702 5 - 5 - 5410 5 - 5

27 \cdot 102 5 - 5 - 27 \cdot 210 5 - 5

Multiply the numbers:

27 \cdot 10 = 270

= 2702 5 - 5 - 27 \cdot 210 5 - 5

Multiply the numbers:

27 \cdot 2 = 54

= 2702 5 - 5 - 5410 5 - 5

= 2702 5 - 5 - 5410 5 - 5

= 2702 5 - 5 - 5410 5 - 5

2 (5 - 5) (5 + 5) = 40

2 (5 - 5) (5 + 5)

Expand

(5 - 5) (5 + 5) : 20

(5 - 5) (5 + 5)

Apply Difference of Two Squares Formula:

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

a = 5, b = 5

= 5^{2} - (5)^{2}

Simplify

5^{2} - (5)^{2} : 20

5^{2} - (5)^{2}

5^{2} = 25

5^{2}

5^{2} = 25

= 25

(5)^{2} = 5

(5)^{2}

Apply radical rule:

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

= (5^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 5^{\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

= 5

= 25 - 5

Subtract the numbers:

25 - 5 = 20

= 20

= 20

= 2 \cdot 20

Expand

2 \cdot 20 : 40

2 \cdot 20

Distribute parentheses

= 2 \cdot 20

Multiply the numbers:

2 \cdot 20 = 40

= 40

= 40

= \frac{270 2 5 - 5 - 54 10 5 - 5}{40}

Factor

2702 5 - 5 - 5410 5 - 5 : 54 5 - 5 (52 - 10)

2702 5 - 5 - 5410 5 - 5

Rewrite as

= 5 \cdot 54 5 - 5 2 - 54 5 - 5 10

Factor out common term

54 5 - 5

= 54 5 - 5 (52 - 10)

= \frac{54 5 - 5 ( 5 2 - 10 )}{40}

Cancel the common factor:

2

= \frac{27 ( 5 2 - 10 ) 5 - 5}{20}

= \frac{27 ( 5 2 - 10 ) 5 - 5}{20}

= \frac{27 ( 5 2 - 10 ) 5 - 5}{20}

Popular Examples

cos^3(5)

cos^{3} (5)

48cos(60)

48 cos (6 0^{\circ})

tan(50)+tan(20)

tan (5 0^{\circ}) + tan (2 0^{\circ})

sin(45)cos(15)+cos(45)sin(15)

sin (4 5^{\circ}) cos (1 5^{\circ}) + cos (4 5^{\circ}) sin (1 5^{\circ})

sin(1380)

sin (138 0^{\circ})

Frequently Asked Questions (FAQ)

What is the value of (27)/(tan(72)) ?
The value of (27)/(tan(72)) is (27(5sqrt(2)-sqrt(10))sqrt(5-\sqrt{5)})/(20)