What is the value of sin((5pi)/(12))cos(pi/(12)) ?

The value of sin((5pi)/(12))cos(pi/(12)) is (2+sqrt(3))/4

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

sin((5pi)/(12))cos(pi/(12))

Solution

sin (\frac{5 π}{12}) cos (\frac{π}{12})

Solution

\frac{2 + 3}{4}

Decimal Notation

0.93301 \dots

Solution steps

sin (\frac{5 π}{12}) cos (\frac{π}{12})

Rewrite using trig identities:

\frac{sin ( \frac{π}{2} ) + sin ( \frac{π}{3} )}{2}

sin (\frac{5 π}{12}) cos (\frac{π}{12})

Use the Product to Sum identity:

sin (s) cos (t) = \frac{1}{2} (sin (s + t) + sin (s - t))

= \frac{1}{2} (sin (\frac{5 π}{12} + \frac{π}{12}) + sin (\frac{5 π}{12} - \frac{π}{12}))

Simplify:

\frac{5 π}{12} + \frac{π}{12} = \frac{π}{2}

\frac{5 π}{12} + \frac{π}{12}

Apply rule

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

= \frac{5 π + π}{12}

Add similar elements:

5 π + π = 6 π

= \frac{6 π}{12}

Cancel the common factor:

6

= \frac{π}{2}

Simplify:

\frac{5 π}{12} - \frac{π}{12} = \frac{π}{3}

\frac{5 π}{12} - \frac{π}{12}

Apply rule

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

= \frac{5 π - π}{12}

Add similar elements:

5 π - π = 4 π

= \frac{4 π}{12}

Cancel the common factor:

4

= \frac{π}{3}

\frac{1}{2} (sin (\frac{π}{2}) + sin (\frac{π}{3})) = \frac{sin ( \frac{π}{2} ) + sin ( \frac{π}{3} )}{2}

\frac{1}{2} (sin (\frac{π}{2}) + sin (\frac{π}{3}))

Multiply fractions:

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

= \frac{1 \cdot ( sin ( \frac{π}{2} ) + sin ( \frac{π}{3} ) )}{2}

1 \cdot (sin (\frac{π}{2}) + sin (\frac{π}{3})) = sin (\frac{π}{2}) + sin (\frac{π}{3})

1 \cdot (sin (\frac{π}{2}) + sin (\frac{π}{3}))

Multiply:

1 \cdot (sin (\frac{π}{2}) + sin (\frac{π}{3})) = (sin (\frac{π}{2}) + sin (\frac{π}{3}))

= (sin (\frac{π}{2}) + sin (\frac{π}{3}))

Remove parentheses:

(a) = a

= sin (\frac{π}{2}) + sin (\frac{π}{3})

= \frac{sin ( \frac{π}{2} ) + sin ( \frac{π}{3} )}{2}

= \frac{sin ( \frac{π}{2} ) + sin ( \frac{π}{3} )}{2}

= \frac{sin ( \frac{π}{2} ) + sin ( \frac{π}{3} )}{2}

Use the following trivial identity:

sin (\frac{π}{2}) = 1

sin (\frac{π}{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}

= 1

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}

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

Simplify

\frac{1 + \frac{3}{2}}{2} : \frac{2 + 3}{4}

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

Join

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

1 + \frac{3}{2}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 + 3}{2}

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 + 3}{4}

= \frac{2 + 3}{4}

Popular Examples

45sin(60)

45 sin (6 0^{\circ})

40sin(20)

40 sin (2 0^{\circ})

8sin(0.3)e^{-0.3}-1

8 sin (0.3) e^{- 0.3} - 1

2tan(pi/4)sec^2(pi/4)

2 tan (\frac{π}{4}) sec^{2} (\frac{π}{4})

2sin(3)

2 sin (3)

Frequently Asked Questions (FAQ)

What is the value of sin((5pi)/(12))cos(pi/(12)) ?
The value of sin((5pi)/(12))cos(pi/(12)) is (2+sqrt(3))/4