What is the value of 6sin(pi/4)cos(pi/6) ?

The value of 6sin(pi/4)cos(pi/6) is (3sqrt(6))/2

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6sin(pi/4)cos(pi/6)

6 sin (\frac{π}{4}) cos (\frac{π}{6})

\frac{3 6}{2}

Decimal Notation

3.67423 \dots

Solution steps

6 sin (\frac{π}{4}) cos (\frac{π}{6})

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{4})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} 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 (\frac{π}{6}) = \frac{3}{2}

cos (\frac{π}{6})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} 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}

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

Simplify

6 \cdot \frac{2}{2} \cdot \frac{3}{2} : \frac{3 6}{2}

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

Multiply fractions:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{6 2 3}{4}

Cancel the common factor:

2

= \frac{3 2 3}{2}

Apply radical rule:

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= \frac{3 3}{2}

Rationalize

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

\frac{3 3}{2}

Multiply by the conjugate

\frac{2}{2}

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

332 = 36

332

Apply radical rule:

a b = a \cdot b

32 = 3 \cdot 2

= 3 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 36

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{3 6}{2}

= \frac{3 6}{2}

= \frac{3 6}{2}

arctan (tan (- \frac{5 π}{3}))

2 - \frac{sin ( 2 )}{cos ( 2 )}

\frac{8}{tan ( 4 0 ^{\circ} )}

arctan (\frac{2.5}{4.75})

- arcsin (\frac{1}{3})

What is the value of 6sin(pi/4)cos(pi/6) ?
The value of 6sin(pi/4)cos(pi/6) is (3sqrt(6))/2