What is the value of (20)/(sin(36)) ?

The value of (20)/(sin(36)) is 2(5sqrt(2)+sqrt(10))sqrt(5-\sqrt{5)}

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

(20)/(sin(36))

Solution

\frac{20}{sin ( 3 6 ^{\circ} )}

Solution

2 (52 + 10) 5 - 5

Decimal Notation

34.02603 \dots

Solution steps

\frac{20}{sin ( 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}

= \frac{20}{\frac{2 5 - 5}{4}}

Simplify

\frac{20}{\frac{2 5 - 5}{4}} : 2 (52 + 10) 5 - 5

\frac{20}{\frac{2 5 - 5}{4}}

Apply the fraction rule:

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

= \frac{20 \cdot 4}{2 5 - 5}

Multiply the numbers:

20 \cdot 4 = 80

= \frac{80}{2 5 - 5}

Factor

80 : 2^{4} \cdot 5

Factor

80 = 2^{4} \cdot 5

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

Cancel

\frac{2 ^{4} \cdot 5}{2 5 - 5} : \frac{5 \cdot 2 ^{\frac{7}{2}}}{5 - 5}

\frac{2 ^{4} \cdot 5}{2 5 - 5}

Apply radical rule:

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

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

Apply exponent rule:

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

\frac{2 ^{4}}{2 ^{\frac{1}{2}}} = 2^{4 - \frac{1}{2}}

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

Subtract the numbers:

4 - \frac{1}{2} = \frac{7}{2}

= \frac{5 \cdot 2 ^{\frac{7}{2}}}{5 - 5}

= \frac{5 \cdot 2 ^{\frac{7}{2}}}{5 - 5}

2^{\frac{7}{2}} = 2^{3} 2

2^{\frac{7}{2}}

2^{\frac{7}{2}} = 2^{3 + \frac{1}{2}}

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

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

= 2^{3} 2

= \frac{5 \cdot 2 ^{3} 2}{5 - 5}

5 \cdot 2^{3} 2 = 402

5 \cdot 2^{3} 2

2^{3} = 8

= 5 \cdot 82

Multiply the numbers:

5 \cdot 8 = 40

= 402

= \frac{40 2}{5 - 5}

Rationalize

\frac{40 2}{5 - 5} : 2 (52 + 10) 5 - 5

\frac{40 2}{5 - 5}

Multiply by the conjugate

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

= \frac{40 2 5 - 5}{5 - 5 5 - 5}

5 - 5 5 - 5 = 5 - 5

5 - 5 5 - 5

Apply radical rule:

a a = a

5 - 5 5 - 5 = 5 - 5

= 5 - 5

= \frac{40 2 5 - 5}{5 - 5}

Factor

5 - 5 : 5 (5 - 1)

5 - 5

5 = 55

= 55 - 5

Factor out common term

5

= 5 (5 - 1)

= \frac{40 2 5 - 5}{5 ( 5 - 1 )}

Factor

40 : 2^{3} \cdot 5

Factor

40 = 2^{3} \cdot 5

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

Cancel

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

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

Apply radical rule:

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

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

Apply exponent rule:

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

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

= \frac{2 ^{3} 2 \cdot 5 ^{- \frac{1}{2} + 1} 5 - 5}{5 - 1}

Subtract the numbers:

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

= \frac{2 ^{3} \cdot 5 ^{\frac{1}{2}} 2 5 - 5}{5 - 1}

Apply radical rule:

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

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

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

Simplify

2^{3} 52 5 - 5 : 2^{3} 10 5 - 5

2^{3} 52 5 - 5

Apply radical rule:

a b = a \cdot b

52 5 - 5 = 5 \cdot 2 (5 - 5)

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

Multiply the numbers:

5 \cdot 2 = 10

= 2^{3} 10 (5 - 5)

Apply radical rule:

assuming

a \geq 0, b \geq 0

10 (5 - 5) = 10 5 - 5

= 2^{3} 10 5 - 5

= \frac{2 ^{3} 10 5 - 5}{5 - 1}

2^{3} = 8

= \frac{8 10 5 - 5}{5 - 1}

Multiply by the conjugate

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

= \frac{8 10 5 - 5 ( 5 + 1 )}{( 5 - 1 ) ( 5 + 1 )}

810 5 - 5 (5 + 1) = 402 5 - 5 + 810 5 - 5

810 5 - 5 (5 + 1)

= 810 (5 + 1) 5 - 5

Apply the distributive law:

a (b + c) = ab + a c

a = 810 5 - 5, b = 5, c = 1

= 810 5 - 5 5 + 810 5 - 5 \cdot 1

= 8105 5 - 5 + 8 \cdot 1 \cdot 10 5 - 5

Simplify

8105 5 - 5 + 8 \cdot 1 \cdot 10 5 - 5 : 402 5 - 5 + 810 5 - 5

8105 5 - 5 + 8 \cdot 1 \cdot 10 5 - 5

8105 5 - 5 = 402 5 - 5

8105 5 - 5

Factor integer

10 = 5 \cdot 2

= 8 5 \cdot 2 5 5 - 5

Apply radical rule:

5 \cdot 2 = 52

= 8525 5 - 5

Apply radical rule:

a a = a

55 = 5

= 8 \cdot 52 5 - 5

Multiply the numbers:

8 \cdot 5 = 40

= 402 5 - 5

8 \cdot 1 \cdot 10 5 - 5 = 810 5 - 5

8 \cdot 1 \cdot 10 5 - 5

Multiply the numbers:

8 \cdot 1 = 8

= 810 5 - 5

= 402 5 - 5 + 810 5 - 5

= 402 5 - 5 + 810 5 - 5

(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{40 2 5 - 5 + 8 10 5 - 5}{4}

Factor

402 5 - 5 + 810 5 - 5 : 8 5 - 5 (52 + 10)

402 5 - 5 + 810 5 - 5

Rewrite as

= 5 \cdot 8 5 - 5 2 + 8 5 - 5 10

Factor out common term

8 5 - 5

= 8 5 - 5 (52 + 10)

= \frac{8 5 - 5 ( 5 2 + 10 )}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2 (52 + 10) 5 - 5

= 2 (52 + 10) 5 - 5

= 2 (52 + 10) 5 - 5

Popular Examples

1/(1+sin(pi/2))

\frac{1}{1 + sin ( \frac{π}{2} )}

1/2 (25)(12)sin(70)

\frac{1}{2} (25) (12) sin (7 0^{\circ})

30*sin(75)

30 \cdot sin (7 5^{\circ})

25*sin(25)

25 \cdot sin (2 5^{\circ})

sin((2pi)/5)cos((2pi)/5)

sin (\frac{2 π}{5}) cos (\frac{2 π}{5})

Frequently Asked Questions (FAQ)

What is the value of (20)/(sin(36)) ?
The value of (20)/(sin(36)) is 2(5sqrt(2)+sqrt(10))sqrt(5-\sqrt{5)}