What is the value of sin(75)sqrt(2) ?

The value of sin(75)sqrt(2) is (sqrt(3)+1)/2

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Popular >

sin(75)sqrt(2)

Solution

sin (7 5^{\circ}) 2

Solution

\frac{3 + 1}{2}

Decimal Notation

1.36602 \dots

Solution steps

sin (7 5^{\circ}) 2

Rewrite using trig identities:

sin (7 5^{\circ}) = \frac{6 + 2}{4}

sin (7 5^{\circ})

Rewrite using trig identities:

sin (4 5^{\circ}) cos (3 0^{\circ}) + cos (4 5^{\circ}) sin (3 0^{\circ})

sin (7 5^{\circ})

Write

sin (7 5^{\circ})

sin (4 5^{\circ} + 3 0^{\circ})

= sin (4 5^{\circ} + 3 0^{\circ})

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (4 5^{\circ}) cos (3 0^{\circ}) + cos (4 5^{\circ}) sin (3 0^{\circ})

= sin (4 5^{\circ}) cos (3 0^{\circ}) + cos (4 5^{\circ}) sin (3 0^{\circ})

Use the following trivial identity:

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

sin (4 5^{\circ})

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} 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 (3 0^{\circ}) = \frac{3}{2}

cos (3 0^{\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{3}{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}

Use the following trivial identity:

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 0^{\circ})

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

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

Simplify

\frac{2}{2} \cdot \frac{3}{2} + \frac{2}{2} \cdot \frac{1}{2} : \frac{6 + 2}{4}

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

\frac{2}{2} \cdot \frac{3}{2} = \frac{6}{4}

\frac{2}{2} \cdot \frac{3}{2}

Multiply fractions:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 3}{4}

Simplify

23 : 6

23

Apply radical rule:

a b = a \cdot b

23 = 2 \cdot 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

= \frac{6}{4}

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

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

Multiply fractions:

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

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

Multiply:

2 \cdot 1 = 2

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{4}

= \frac{6}{4} + \frac{2}{4}

Apply rule

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

= \frac{6 + 2}{4}

= \frac{6 + 2}{4}

= \frac{6 + 2}{4} 2

Simplify

\frac{6 + 2}{4} 2 : \frac{3 + 1}{2}

\frac{6 + 2}{4} 2

Multiply fractions:

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

= \frac{( 6 + 2 ) 2}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{2 ( 6 + 2 )}{2 ^{2}}

Cancel

\frac{( 6 + 2 ) 2}{2 ^{2}} : \frac{6 + 2}{2 ^{\frac{3}{2}}}

\frac{( 6 + 2 ) 2}{2 ^{2}}

Apply radical rule:

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

= \frac{2 ^{\frac{1}{2}} ( 6 + 2 )}{2 ^{2}}

Apply exponent rule:

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

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

= \frac{6 + 2}{2 ^{2 - \frac{1}{2}}}

Subtract the numbers:

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

= \frac{6 + 2}{2 ^{\frac{3}{2}}}

= \frac{6 + 2}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

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

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

Apply exponent rule:

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

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

Refine

= 22

= \frac{6 + 2}{2 2}

Rationalize

\frac{6 + 2}{2 2} : \frac{1 + 3}{2}

\frac{6 + 2}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{( 6 + 2 ) 2}{2 2 2}

(6 + 2) 2 = 23 + 2

(6 + 2) 2

= 2 (6 + 2)

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = 6, c = 2

= 26 + 22

Simplify

26 + 22 : 23 + 2

26 + 22

26 = 23

26

Factor integer

6 = 2 \cdot 3

= 2 2 \cdot 3

Apply radical rule:

2 \cdot 3 = 23

= 223

Apply radical rule:

a a = a

22 = 2

= 23

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= 23 + 2

= 23 + 2

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

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

Add similar elements:

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

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

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

2 \cdot \frac{1}{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^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

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

Factor

23 + 2 : 2 (3 + 1)

23 + 2

Rewrite as

= 23 + 2 \cdot 1

Factor out common term

2

= 2 (3 + 1)

= \frac{2 ( 3 + 1 )}{4}

Cancel the common factor:

2

= \frac{3 + 1}{2}

= \frac{3 + 1}{2}

= \frac{3 + 1}{2}

Popular Examples

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csc (\frac{13 π}{3})

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4.9 \cdot 1 0^{6} \cdot 1.6 \cdot 1 0^{- 19} \cdot 0.19 \cdot sin (6 4^{\circ})

cot(28)

cot (2 8^{\circ})

arctan(-0.2)

arctan (- 0.2)

5*cos(45)

5 \cdot cos (4 5^{\circ})

Frequently Asked Questions (FAQ)

What is the value of sin(75)sqrt(2) ?
The value of sin(75)sqrt(2) is (sqrt(3)+1)/2