What is the value of cos(pi/6)-sin((3pi)/4) ?

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

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cos(pi/6)-sin((3pi)/4)

cos (\frac{π}{6}) - sin (\frac{3 π}{4})

\frac{3 - 2}{2}

Decimal Notation

0.15891 \dots

Solution steps

cos (\frac{π}{6}) - sin (\frac{3 π}{4})

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}

Use the following trivial identity:

sin (\frac{3 π}{4}) = \frac{2}{2}

sin (\frac{3 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}

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

Simplify

\frac{3}{2} - \frac{2}{2} : \frac{3 - 2}{2}

\frac{3}{2} - \frac{2}{2}

Apply rule

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

= \frac{3 - 2}{2}

= \frac{3 - 2}{2}

- 1 - arctan (- 1)

60 cos (7 5^{\circ})

cos^{2} (\frac{3 π}{8})

cot (arcsec (\frac{5}{3}) + arcsin (\frac{5}{13}))

arccos (\frac{4}{5}) + arctan (\frac{1}{7})

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