What is the value of -4cos((7pi)/(12))cos((5pi)/(12)) ?

The value of -4cos((7pi)/(12))cos((5pi)/(12)) is 2-sqrt(3)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

-4cos((7pi)/(12))cos((5pi)/(12))

Solution

- 4 cos (\frac{7 π}{12}) cos (\frac{5 π}{12})

Solution

2 - 3

Decimal Notation

0.26794 \dots

Solution steps

- 4 cos (\frac{7 π}{12}) cos (\frac{5 π}{12})

Rewrite using trig identities:

4 cos (\frac{7 π}{12}) cos (\frac{5 π}{12}) = 2 (cos (π) + cos (\frac{π}{6}))

4 cos (\frac{7 π}{12}) cos (\frac{5 π}{12})

Use the Product to Sum identity:

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

= 4 \cdot \frac{1}{2} (cos (\frac{7 π}{12} - \frac{5 π}{12}) + cos (\frac{7 π}{12} + \frac{5 π}{12}))

Simplify:

\frac{7 π}{12} - \frac{5 π}{12} = \frac{π}{6}

\frac{7 π}{12} - \frac{5 π}{12}

Apply rule

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

= \frac{7 π - 5 π}{12}

Add similar elements:

7 π - 5 π = 2 π

= \frac{2 π}{12}

Cancel the common factor:

2

= \frac{π}{6}

Simplify:

\frac{7 π}{12} + \frac{5 π}{12} = π

\frac{7 π}{12} + \frac{5 π}{12}

Apply rule

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

= \frac{7 π + 5 π}{12}

Add similar elements:

7 π + 5 π = 12 π

= \frac{12 π}{12}

Divide the numbers:

\frac{12}{12} = 1

= π

4 \cdot \frac{1}{2} (cos (\frac{π}{6}) + cos (π)) = 2 (cos (π) + cos (\frac{π}{6}))

4 \cdot \frac{1}{2} (cos (\frac{π}{6}) + cos (π))

Multiply fractions:

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

= \frac{1 \cdot 4 ( cos ( \frac{π}{6} ) + cos ( π ) )}{2}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4 ( cos ( π ) + cos ( \frac{π}{6} ) )}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 (cos (π) + cos (\frac{π}{6}))

= 2 (cos (π) + cos (\frac{π}{6}))

= - 2 (cos (π) + cos (\frac{π}{6}))

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

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}

= (- 1)

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}

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

Simplify

- 2 (- 1 + \frac{3}{2}) : 2 - 3

- 2 (- 1 + \frac{3}{2})

Apply the distributive law:

a (b + c) = ab + a c

a = - 2, b = - 1, c = \frac{3}{2}

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

Apply minus-plus rules

- (- a) = a, + (- a) = - a

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

Simplify

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

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

2 \cdot 1 = 2

2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 2

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

2 \cdot \frac{3}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 3

= 2 - 3

= 2 - 3

= 2 - 3

Popular Examples

sec(1.24)

sec (1.24)

cos(162)

cos (16 2^{\circ})

arctan((-3)/1)

arctan (\frac{- 3}{1})

arccos(15)

arccos (15)

cos^2(15)+sin^2(15)

cos^{2} (1 5^{\circ}) + sin^{2} (1 5^{\circ})

Frequently Asked Questions (FAQ)

What is the value of -4cos((7pi)/(12))cos((5pi)/(12)) ?
The value of -4cos((7pi)/(12))cos((5pi)/(12)) is 2-sqrt(3)