What is the value of 4(sin(15)cos^3(15)-sin^3(15)cos(15)) ?

The value of 4(sin(15)cos^3(15)-sin^3(15)cos(15)) is (sqrt(3))/2

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4(sin(15)cos^3(15)-sin^3(15)cos(15))

Solution

4 (sin (1 5^{\circ}) cos^{3} (1 5^{\circ}) - sin^{3} (1 5^{\circ}) cos (1 5^{\circ}))

Solution

\frac{3}{2}

Decimal Notation

0.86602 \dots

Solution steps

4 (sin (1 5^{\circ}) cos^{3} (1 5^{\circ}) - sin^{3} (1 5^{\circ}) cos (1 5^{\circ}))

Rewrite using trig identities:

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

sin (1 5^{\circ})

Rewrite using trig identities:

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

sin (1 5^{\circ})

Write

sin (1 5^{\circ})

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

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

Use the Angle Difference 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}

Rewrite using trig identities:

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

cos (1 5^{\circ})

Rewrite using trig identities:

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

cos (1 5^{\circ})

Write

cos (1 5^{\circ})

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

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

Use the Angle Difference identity:

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

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

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

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:

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:

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:

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}

= 4 \frac{6 - 2}{4} (\frac{6 + 2}{4})^{3} - (\frac{6 - 2}{4})^{3} \frac{6 + 2}{4}

Simplify

4 \frac{6 - 2}{4} (\frac{6 + 2}{4})^{3} - (\frac{6 - 2}{4})^{3} \frac{6 + 2}{4} : \frac{3}{2}

4 \frac{6 - 2}{4} (\frac{6 + 2}{4})^{3} - (\frac{6 - 2}{4})^{3} \frac{6 + 2}{4}

\frac{6 - 2}{4} (\frac{6 + 2}{4})^{3} = \frac{2 + 3}{16}

\frac{6 - 2}{4} (\frac{6 + 2}{4})^{3}

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

(\frac{6 + 2}{4})^{3}

Apply exponent rule:

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

= \frac{( 6 + 2 ) ^{3}}{4 ^{3}}

(6 + 2)^{3} = 126 + 202

(6 + 2)^{3}

Apply Perfect Cube Formula:

(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

a = 6, b = 2

= (6)^{3} + 3 (6)^{2} 2 + 36 (2)^{2} + (2)^{3}

Simplify

(6)^{3} + 3 (6)^{2} 2 + 36 (2)^{2} + (2)^{3} : 126 + 202

(6)^{3} + 3 (6)^{2} 2 + 36 (2)^{2} + (2)^{3}

(6)^{3} = 66

(6)^{3}

Apply radical rule:

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

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

Apply exponent rule:

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

= 6^{\frac{1}{2} \cdot 3}

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

\frac{1}{2} \cdot 3

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{2}

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

6^{\frac{3}{2}} = 66

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

6^{\frac{3}{2}} = 6^{1 + \frac{1}{2}}

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

Apply exponent rule:

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

= 6^{1} \cdot 6^{\frac{1}{2}}

Refine

= 66

= 66

3 (6)^{2} 2 = 182

3 (6)^{2} 2

(6)^{2} = 6

(6)^{2}

Apply radical rule:

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

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

Apply exponent rule:

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

= 6^{\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

= 6

= 3 \cdot 62

Multiply the numbers:

3 \cdot 6 = 18

= 182

36 (2)^{2} = 66

36 (2)^{2}

(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

= 3 \cdot 26

Multiply the numbers:

3 \cdot 2 = 6

= 66

(2)^{3} = 22

(2)^{3}

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

\frac{1}{2} \cdot 3

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{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

= 22

= 66 + 182 + 66 + 22

Add similar elements:

182 + 22 = 202

= 66 + 202 + 66

Add similar elements:

66 + 66 = 126

= 126 + 202

= 126 + 202

= \frac{12 6 + 20 2}{4 ^{3}}

Factor

126 + 202 : 4 (36 + 52)

126 + 202

Rewrite as

= 4 \cdot 36 + 4 \cdot 52

Factor out common term

4

= 4 (36 + 52)

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

Cancel the common factor:

4

= \frac{3 6 + 5 2}{4 ^{2}}

= \frac{6 - 2}{4} \cdot \frac{3 6 + 5 2}{4 ^{2}}

Multiply fractions:

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

= \frac{( 6 - 2 ) ( 3 6 + 5 2 )}{4 \cdot 4 ^{2}}

4 \cdot 4^{2} = 4^{3}

4 \cdot 4^{2}

Apply exponent rule:

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

4 \cdot 4^{2} = 4^{1 + 2}

= 4^{1 + 2}

Add the numbers:

1 + 2 = 3

= 4^{3}

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

4^{3} = 64

= \frac{( 6 - 2 ) ( 3 6 + 5 2 )}{64}

Expand

(6 - 2) (36 + 52) : 8 + 43

(6 - 2) (36 + 52)

Apply FOIL method:

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

a = 6, b = - 2, c = 36, d = 52

= 6 \cdot 36 + 6 \cdot 52 + (- 2) \cdot 36 + (- 2) \cdot 52

Apply minus-plus rules

+ (- a) = - a

= 366 + 562 - 326 - 522

Simplify

366 + 562 - 326 - 522 : 8 + 43

366 + 562 - 326 - 522

Add similar elements:

562 - 326 = 226

= 366 + 226 - 522

366 = 18

366

Apply radical rule:

a a = a

66 = 6

= 3 \cdot 6

Multiply the numbers:

3 \cdot 6 = 18

= 18

226 = 43

226

Factor integer

6 = 2 \cdot 3

= 22 2 \cdot 3

Apply radical rule:

2 \cdot 3 = 23

= 2223

Apply radical rule:

a a = a

22 = 2

= 2 \cdot 23

Multiply the numbers:

2 \cdot 2 = 4

= 43

522 = 10

522

Apply radical rule:

a a = a

22 = 2

= 5 \cdot 2

Multiply the numbers:

5 \cdot 2 = 10

= 10

= 18 + 43 - 10

Subtract the numbers:

18 - 10 = 8

= 8 + 43

= 8 + 43

= \frac{8 + 4 3}{64}

Factor

8 + 43 : 4 (2 + 3)

8 + 43

Rewrite as

= 4 \cdot 2 + 43

Factor out common term

4

= 4 (2 + 3)

= \frac{4 ( 2 + 3 )}{64}

Cancel the common factor:

4

= \frac{2 + 3}{16}

(\frac{6 - 2}{4})^{3} \frac{6 + 2}{4} = \frac{2 - 3}{16}

(\frac{6 - 2}{4})^{3} \frac{6 + 2}{4}

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

(\frac{6 - 2}{4})^{3}

Apply exponent rule:

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

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

(6 - 2)^{3} = 126 - 202

(6 - 2)^{3}

Apply Perfect Cube Formula:

(a - b)^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}

a = 6, b = 2

= (6)^{3} - 3 (6)^{2} 2 + 36 (2)^{2} - (2)^{3}

Simplify

(6)^{3} - 3 (6)^{2} 2 + 36 (2)^{2} - (2)^{3} : 126 - 202

(6)^{3} - 3 (6)^{2} 2 + 36 (2)^{2} - (2)^{3}

(6)^{3} = 66

(6)^{3}

Apply radical rule:

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

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

Apply exponent rule:

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

= 6^{\frac{1}{2} \cdot 3}

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

\frac{1}{2} \cdot 3

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{2}

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

6^{\frac{3}{2}} = 66

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

6^{\frac{3}{2}} = 6^{1 + \frac{1}{2}}

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

Apply exponent rule:

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

= 6^{1} \cdot 6^{\frac{1}{2}}

Refine

= 66

= 66

3 (6)^{2} 2 = 182

3 (6)^{2} 2

(6)^{2} = 6

(6)^{2}

Apply radical rule:

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

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

Apply exponent rule:

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

= 6^{\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

= 6

= 3 \cdot 62

Multiply the numbers:

3 \cdot 6 = 18

= 182

36 (2)^{2} = 66

36 (2)^{2}

(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

= 3 \cdot 26

Multiply the numbers:

3 \cdot 2 = 6

= 66

(2)^{3} = 22

(2)^{3}

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

\frac{1}{2} \cdot 3

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{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

= 22

= 66 - 182 + 66 - 22

Add similar elements:

- 182 - 22 = - 202

= 66 - 202 + 66

Add similar elements:

66 + 66 = 126

= 126 - 202

= 126 - 202

= \frac{12 6 - 20 2}{4 ^{3}}

Factor

126 - 202 : 4 (36 - 52)

126 - 202

Rewrite as

= 4 \cdot 36 - 4 \cdot 52

Factor out common term

4

= 4 (36 - 52)

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

Cancel the common factor:

4

= \frac{3 6 - 5 2}{4 ^{2}}

= \frac{6 + 2}{4} \cdot \frac{3 6 - 5 2}{4 ^{2}}

Multiply fractions:

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

= \frac{( 3 6 - 5 2 ) ( 6 + 2 )}{4 ^{2} \cdot 4}

4^{2} \cdot 4 = 4^{3}

4^{2} \cdot 4

Apply exponent rule:

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

4^{2} \cdot 4 = 4^{2 + 1}

= 4^{2 + 1}

Add the numbers:

2 + 1 = 3

= 4^{3}

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

4^{3} = 64

= \frac{( 3 6 - 5 2 ) ( 6 + 2 )}{64}

Expand

(36 - 52) (6 + 2) : 8 - 43

(36 - 52) (6 + 2)

Apply FOIL method:

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

a = 36, b = - 52, c = 6, d = 2

= 366 + 362 + (- 52) 6 + (- 52) 2

Apply minus-plus rules

+ (- a) = - a

= 366 + 362 - 526 - 522

Simplify

366 + 362 - 526 - 522 : 8 - 43

366 + 362 - 526 - 522

Add similar elements:

362 - 526 = - 226

= 366 - 226 - 522

366 = 18

366

Apply radical rule:

a a = a

66 = 6

= 3 \cdot 6

Multiply the numbers:

3 \cdot 6 = 18

= 18

226 = 43

226

Factor integer

6 = 2 \cdot 3

= 22 2 \cdot 3

Apply radical rule:

2 \cdot 3 = 23

= 2223

Apply radical rule:

a a = a

22 = 2

= 2 \cdot 23

Multiply the numbers:

2 \cdot 2 = 4

= 43

522 = 10

522

Apply radical rule:

a a = a

22 = 2

= 5 \cdot 2

Multiply the numbers:

5 \cdot 2 = 10

= 10

= 18 - 43 - 10

Subtract the numbers:

18 - 10 = 8

= 8 - 43

= 8 - 43

= \frac{8 - 4 3}{64}

Factor

8 - 43 : 4 (2 - 3)

8 - 43

Rewrite as

= 4 \cdot 2 - 43

Factor out common term

4

= 4 (2 - 3)

= \frac{4 ( 2 - 3 )}{64}

Cancel the common factor:

4

= \frac{2 - 3}{16}

= 4 (\frac{2 + 3}{16} - \frac{2 - 3}{16})

Simplify

\frac{2 + 3}{16} - \frac{2 - 3}{16} : \frac{- ( 2 - 3 ) + 2 + 3}{16}

\frac{2 + 3}{16} - \frac{2 - 3}{16}

Apply rule

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

= \frac{2 + 3 - ( 2 - 3 )}{16}

= 4 \cdot \frac{- ( 2 - 3 ) + 2 + 3}{16}

\frac{2 + 3 - ( 2 - 3 )}{16} = \frac{3}{8}

\frac{2 + 3 - ( 2 - 3 )}{16}

Expand

2 + 3 - (2 - 3) : 23

2 + 3 - (2 - 3)

- (2 - 3) : - 2 + 3

- (2 - 3)

Distribute parentheses

= - (2) - (- 3)

Apply minus-plus rules

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

= - 2 + 3

= 2 + 3 - 2 + 3

Simplify

2 + 3 - 2 + 3 : 23

2 + 3 - 2 + 3

Add similar elements:

3 + 3 = 23

= 2 + 23 - 2

2 - 2 = 0

= 23

= 23

= \frac{2 3}{16}

Cancel the common factor:

2

= \frac{3}{8}

= 4 \cdot \frac{3}{8}

Multiply fractions:

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

= \frac{3 \cdot 4}{8}

Cancel the common factor:

4

= \frac{3}{2}

= \frac{3}{2}

Popular Examples

cos(pi/4-(7pi)/6)

cos (\frac{π}{4} - \frac{7 π}{6})

arccos(634148)

arccos (634148)

tan(45)-tan(30)

tan (4 5^{\circ}) - tan (3 0^{\circ})

-arctan((sqrt(3))/3)

- arctan (\frac{3}{3})

cot(78)

cot (7 8^{\circ})

Frequently Asked Questions (FAQ)

What is the value of 4(sin(15)cos^3(15)-sin^3(15)cos(15)) ?
The value of 4(sin(15)cos^3(15)-sin^3(15)cos(15)) is (sqrt(3))/2