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

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

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cos(pi/3)cos(pi/6)+sin(pi/3)sin(pi/6)

cos (\frac{π}{3}) cos (\frac{π}{6}) + sin (\frac{π}{3}) sin (\frac{π}{6})

\frac{3}{2}

Decimal Notation

0.86602 \dots

Solution steps

cos (\frac{π}{3}) cos (\frac{π}{6}) + sin (\frac{π}{3}) sin (\frac{π}{6})

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

cos (\frac{π}{3})

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{1}{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}

Use the following trivial identity:

sin (\frac{π}{3}) = \frac{3}{2}

sin (\frac{π}{3})

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{3}{2}

Use the following trivial identity:

sin (\frac{π}{6}) = \frac{1}{2}

sin (\frac{π}{6})

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}

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

Simplify

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

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

Add similar elements:

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

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

Cancel the common factor:

2

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

Multiply:

1 \cdot 3 = 3

= \frac{3}{2}

= \frac{3}{2}

10 sin (10 0^{\circ})

27 cos (1 8^{\circ})

13 \cdot cos (6 5^{\circ})

sin^{2} (3 5^{\circ}) + sin^{2} (5 5^{\circ})

sin (\frac{18 π}{3})

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