What is the value of sqrt(2)csc(-765)+2csc(742.5) ?

The value of sqrt(2)csc(-765)+2csc(742.5) is -2+2(2sqrt(2-\sqrt{2)}+sqrt(2)sqrt(2-\sqrt{2)})

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sqrt(2)csc(-765)+2csc(742.5)

Solution

2 csc (- 76 5^{\circ}) + 2 csc (742. 5^{\circ})

Solution

- 2 + 2 (2 2 - 2 + 2 2 - 2)

Decimal Notation

3.22625 \dots

Solution steps

2 csc (- 76 5^{\circ}) + 2 csc (742. 5^{\circ})

Use the following property:

csc (- x) = - csc (x)

csc (- 76 5^{\circ}) = - csc (76 5^{\circ})

= 2 (- csc (76 5^{\circ})) + 2 csc (742. 5^{\circ})

csc (76 5^{\circ}) = csc (4 5^{\circ})

csc (76 5^{\circ})

Rewrite

76 5^{\circ}

36 0^{\circ} \cdot 2 + 4 5^{\circ}

= csc (36 0^{\circ} 2 + 4 5^{\circ})

Apply the periodicity of

csc

csc (x + 36 0^{\circ} \cdot k) = csc (x)

csc (36 0^{\circ} \cdot 2 + 4 5^{\circ}) = csc (4 5^{\circ})

= csc (4 5^{\circ})

= 2 (- csc (4 5^{\circ})) + 2 csc (742. 5^{\circ})

csc (742. 5^{\circ}) = csc (22. 5^{\circ})

csc (742. 5^{\circ})

Rewrite

742. 5^{\circ}

36 0^{\circ} \cdot 2 + 22. 5^{\circ}

= csc (36 0^{\circ} 2 + 22. 5^{\circ})

Apply the periodicity of

csc

csc (x + 36 0^{\circ} \cdot k) = csc (x)

csc (36 0^{\circ} \cdot 2 + 22. 5^{\circ}) = csc (22. 5^{\circ})

= csc (22. 5^{\circ})

= 2 (- csc (4 5^{\circ})) + 2 csc (22. 5^{\circ})

Simplify

= - 2 csc (4 5^{\circ}) + 2 csc (22. 5^{\circ})

Use the following trivial identity:

csc (4 5^{\circ}) = 2

csc (4 5^{\circ})

csc (x)

periodicity table with

36 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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

= 2

Rewrite using trig identities:

csc (22. 5^{\circ}) = 2 2 - 2 + 2 2 - 2

csc (22. 5^{\circ})

Rewrite using trig identities:

\frac{1}{sin ( 22. 5 ^{\circ} )}

csc (22. 5^{\circ})

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

= \frac{1}{sin ( 22. 5 ^{\circ} )}

= \frac{1}{sin ( 22. 5 ^{\circ} )}

Rewrite using trig identities:

sin (22. 5^{\circ}) = \frac{2 - 2}{2}

sin (22. 5^{\circ})

Rewrite using trig identities:

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

sin (22. 5^{\circ})

Write

sin (22. 5^{\circ})

sin (\frac{4 5 ^{\circ}}{2})

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

Use the Half Angle identity:

sin (\frac{θ}{2}) = \frac{1 - cos ( θ )}{2}

Use the Double Angle identity

cos (2 θ) = 1 - 2 sin^{2} (θ)

Substitute

θ

with

\frac{θ}{2}

cos (θ) = 1 - 2 sin^{2} (\frac{θ}{2})

Switch sides

2 sin^{2} (\frac{θ}{2}) = 1 - cos (θ)

Divide both sides by

2

sin^{2} (\frac{θ}{2}) = \frac{( 1 - cos ( θ ) )}{2}

Square root both sides

Choose the root sign according to the quadrant of

\frac{θ}{2}

range [0, 9 0^{\circ}] [9 0^{\circ}, 18 0^{\circ}] [18 0^{\circ}, 27 0^{\circ}] [27 0^{\circ}, 36 0^{\circ}] quadrant I II III I V sin positive positive negative negative cos positive negative negative positive

sin (\frac{θ}{2}) = \frac{( 1 - cos ( θ ) )}{2}

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

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

Use the following trivial identity:

cos (4 5^{\circ}) = \frac{2}{2}

cos (4 5^{\circ})

cos (x)

periodicity table with

36 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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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{1 - \frac{2}{2}}{2}

Simplify

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

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

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

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

Join

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

1 - \frac{2}{2}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 - 2}{2}

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 - 2}{4}

= \frac{2 - 2}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{2 - 2}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2 - 2}{2}

= \frac{2 - 2}{2}

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

Simplify

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

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

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{2}{2 - 2}

Rationalize

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

\frac{2}{2 - 2}

Multiply by the conjugate

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

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

2 - 2 2 - 2 = 2 - 2

2 - 2 2 - 2

Apply radical rule:

a a = a

2 - 2 2 - 2 = 2 - 2

= 2 - 2

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

Factor

2 - 2 : 2 (2 - 1)

2 - 2

2 = 22

= 22 - 2

Factor out common term

2

= 2 (2 - 1)

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

Cancel

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

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

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

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

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

Multiply by the conjugate

\frac{2 + 1}{2 + 1}

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

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

2 2 - 2 (2 + 1)

= 2 (2 + 1) 2 - 2

Apply the distributive law:

a (b + c) = ab + a c

a = 2 2 - 2, b = 2, c = 1

= 2 2 - 2 2 + 2 2 - 2 \cdot 1

= 22 2 - 2 + 1 \cdot 2 2 - 2

Simplify

22 2 - 2 + 1 \cdot 2 2 - 2 : 2 2 - 2 + 2 2 - 2

22 2 - 2 + 1 \cdot 2 2 - 2

Apply radical rule:

a a = a

22 = 2

= 2 2 - 2 + 1 \cdot 2 2 - 2

Multiply:

1 \cdot 2 = 2

= 2 2 - 2 + 2 2 - 2

= 2 2 - 2 + 2 2 - 2

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

(2 - 1) (2 + 1)

Apply Difference of Two Squares Formula:

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

a = 2, b = 1

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

Simplify

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

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

Apply rule

1^{a} = 1

1^{2} = 1

= (2)^{2} - 1

(2)^{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

= 2 - 1

Subtract the numbers:

2 - 1 = 1

= 1

= 1

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

Apply rule

\frac{a}{1} = a

= 2 2 - 2 + 2 2 - 2

= 2 2 - 2 + 2 2 - 2

= 2 2 - 2 + 2 2 - 2

= - 22 + 2 (2 2 - 2 + 2 2 - 2)

Simplify

- 22 + 2 (2 2 - 2 + 2 2 - 2) : - 2 + 2 (2 2 - 2 + 2 2 - 2)

- 22 + 2 (2 2 - 2 + 2 2 - 2)

Apply radical rule:

a a = a

22 = 2

= - 2 + 2 (2 2 - 2 + 2 2 - 2)

= - 2 + 2 (2 2 - 2 + 2 2 - 2)

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Frequently Asked Questions (FAQ)

What is the value of sqrt(2)csc(-765)+2csc(742.5) ?
The value of sqrt(2)csc(-765)+2csc(742.5) is -2+2(2sqrt(2-\sqrt{2)}+sqrt(2)sqrt(2-\sqrt{2)})