What is the value of cos((7pi)/(24))sin(pi/(24)) ?

The value of cos((7pi)/(24))sin(pi/(24)) is (sqrt(3)-sqrt(2))/4

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cos((7pi)/(24))sin(pi/(24))

Solution

cos (\frac{7 π}{24}) sin (\frac{π}{24})

Solution

\frac{3 - 2}{4}

Decimal Notation

0.07945 \dots

Solution steps

cos (\frac{7 π}{24}) sin (\frac{π}{24})

Rewrite using trig identities:

\frac{sin ( \frac{π}{3} ) - sin ( \frac{π}{4} )}{2}

cos (\frac{7 π}{24}) sin (\frac{π}{24})

Use the Product to Sum identity:

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

= \frac{1}{2} (sin (\frac{π}{24} + \frac{7 π}{24}) + sin (\frac{π}{24} - \frac{7 π}{24}))

Simplify:

\frac{π}{24} + \frac{7 π}{24} = \frac{π}{3}

\frac{π}{24} + \frac{7 π}{24}

Apply rule

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

= \frac{π + 7 π}{24}

Add similar elements:

π + 7 π = 8 π

= \frac{8 π}{24}

Cancel the common factor:

8

= \frac{π}{3}

Simplify:

\frac{π}{24} - \frac{7 π}{24} = - \frac{π}{4}

\frac{π}{24} - \frac{7 π}{24}

Apply rule

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

= \frac{π - 7 π}{24}

Add similar elements:

π - 7 π = - 6 π

= \frac{- 6 π}{24}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{6 π}{24}

Cancel the common factor:

6

= - \frac{π}{4}

\frac{1}{2} (sin (\frac{π}{3}) + sin (- \frac{π}{4})) = \frac{sin ( \frac{π}{3} ) + sin ( - \frac{π}{4} )}{2}

\frac{1}{2} (sin (\frac{π}{3}) + sin (- \frac{π}{4}))

Multiply fractions:

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

= \frac{1 \cdot ( sin ( \frac{π}{3} ) + sin ( - \frac{π}{4} ) )}{2}

1 \cdot (sin (\frac{π}{3}) + sin (- \frac{π}{4})) = sin (\frac{π}{3}) + sin (- \frac{π}{4})

1 \cdot (sin (\frac{π}{3}) + sin (- \frac{π}{4}))

Multiply:

1 \cdot (sin (\frac{π}{3}) + sin (- \frac{π}{4})) = (sin (\frac{π}{3}) + sin (- \frac{π}{4}))

= (sin (\frac{π}{3}) + sin (- \frac{π}{4}))

Remove parentheses:

(a) = a

= sin (\frac{π}{3}) + sin (- \frac{π}{4})

= \frac{sin ( \frac{π}{3} ) + sin ( - \frac{π}{4} )}{2}

= \frac{sin ( \frac{π}{3} ) + sin ( - \frac{π}{4} )}{2}

Use the following property:

sin (- x) = - sin (x)

sin (- \frac{π}{4}) = - sin (\frac{π}{4})

= \frac{sin ( \frac{π}{3} ) - sin ( \frac{π}{4} )}{2}

= \frac{sin ( \frac{π}{3} ) - sin ( \frac{π}{4} )}{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}{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}

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

Simplify

\frac{\frac{3}{2} - \frac{2}{2}}{2} : \frac{3 - 2}{4}

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

Combine the fractions

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

Apply rule

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

= \frac{3 - 2}{2}

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 - 2}{4}

= \frac{3 - 2}{4}

Popular Examples

e^{cos(pi/2)}

e^{c o s (\frac{π}{2})}

tan(945)

tan (94 5^{\circ})

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sec^2(10)

sec^{2} (1 0^{\circ})

sec(17)

sec (1 7^{\circ})

Frequently Asked Questions (FAQ)

What is the value of cos((7pi)/(24))sin(pi/(24)) ?
The value of cos((7pi)/(24))sin(pi/(24)) is (sqrt(3)-sqrt(2))/4