Is 6sin(pi/6)=2+2sin(pi/6) ?

The answer to whether 6sin(pi/6)=2+2sin(pi/6) is True

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prove 6sin(pi/6)=2+2sin(pi/6)

prove

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

True

Solution steps

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

Manipulating left side

6 sin (\frac{π}{6})

Simplify

6 sin (\frac{π}{6}) : 3

6 sin (\frac{π}{6})

Simplify

sin (\frac{π}{6}) : \frac{1}{2}

sin (\frac{π}{6})

Use the following trivial identity:

sin (\frac{π}{6}) = \frac{1}{2}

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{1}{2}

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

Simplify

6 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{6}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3

= 3

= 3

Manipulating right side

2 + 2 sin (\frac{π}{6})

Simplify

2 + 2 sin (\frac{π}{6}) : 3

2 + 2 sin (\frac{π}{6})

2 sin (\frac{π}{6}) = 1

2 sin (\frac{π}{6})

Simplify

sin (\frac{π}{6}) : \frac{1}{2}

sin (\frac{π}{6})

Use the following trivial identity:

sin (\frac{π}{6}) = \frac{1}{2}

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{1}{2}

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

Multiply fractions:

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

= \frac{2}{2}

Cancel the common factor:

2

= 1

= 2 + 1

Simplify

= 3

= 3

We showed that the two sides could take the same form

\Rightarrow True

\frac{cos ( x + y )}{cos ( x - y )} = \frac{1 - tan ( x ) tan ( y )}{1 + tan ( x ) tan ( y )}

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

cos^{4} (x) - sin^{4} (x) + 1 = 2 cos^{2} (x)

sin (x) cos (x) = \frac{sin ( 2 x )}{2}

- tan (y) + sec (y) = \frac{cos ( y )}{1 + sin ( y )}

Is 6sin(pi/6)=2+2sin(pi/6) ?
The answer to whether 6sin(pi/6)=2+2sin(pi/6) is True