What is the value of 2-4sin^2((5pi)/(12)) ?

The value of 2-4sin^2((5pi)/(12)) is -sqrt(3)

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2-4sin^2((5pi)/(12))

Solution

2 - 4 sin^{2} (\frac{5 π}{12})

Solution

- 3

Decimal Notation

- 1.73205 \dots

Solution steps

2 - 4 sin^{2} (\frac{5 π}{12})

Rewrite using trig identities:

sin^{2} (\frac{5 π}{12}) = \frac{1 - cos ( \frac{5 π}{6} )}{2}

sin^{2} (\frac{5 π}{12})

Use the following identity:

sin^{2} (θ) = \frac{1 - cos ( 2 θ )}{2}

Use the Double Angle identity

cos (2 θ) = 1 - 2 sin^{2} (θ)

Switch sides

2 sin^{2} (θ) - 1 = - cos (2 θ)

Add

1

to both sides

2 sin^{2} (θ) = 1 - cos (2 θ)

Divide both sides by

2

sin^{2} (θ) = \frac{1 - cos ( 2 θ )}{2}

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

Simplify:

2 \cdot \frac{5 π}{12} = \frac{5 π}{6}

2 \cdot \frac{5 π}{12}

Multiply fractions:

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

= \frac{5 π 2}{12}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10 π}{12}

Cancel the common factor:

2

= \frac{5 π}{6}

= \frac{1 - cos ( \frac{5 π}{6} )}{2}

= 2 - 4 \cdot \frac{1 - cos ( \frac{5 π}{6} )}{2}

2 - 4 \cdot \frac{1 - cos ( \frac{5 π}{6} )}{2} = 2 cos (\frac{5 π}{6})

2 - 4 \cdot \frac{1 - cos ( \frac{5 π}{6} )}{2}

4 \cdot \frac{1 - cos ( \frac{5 π}{6} )}{2} = 2 (- cos (\frac{5 π}{6}) + 1)

4 \cdot \frac{1 - cos ( \frac{5 π}{6} )}{2}

Multiply fractions:

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

= \frac{( 1 - cos ( \frac{5 π}{6} ) ) \cdot 4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 (- cos (\frac{5 π}{6}) + 1)

= 2 - 2 (- cos (\frac{5 π}{6}) + 1)

Expand

- 2 (- cos (\frac{5 π}{6}) + 1) : 2 cos (\frac{5 π}{6}) - 2

- 2 (- cos (\frac{5 π}{6}) + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = - 2, b = - cos (\frac{5 π}{6}), c = 1

= - 2 (- cos (\frac{5 π}{6})) + (- 2) \cdot 1

Apply minus-plus rules

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

= 2 cos (\frac{5 π}{6}) - 2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 2 cos (\frac{5 π}{6}) - 2

= 2 + 2 cos (\frac{5 π}{6}) - 2

2 - 2 = 0

= 2 cos (\frac{5 π}{6})

= 2 cos (\frac{5 π}{6})

Use the following trivial identity:

cos (\frac{5 π}{6}) = - \frac{3}{2}

cos (\frac{5 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 (- \frac{3}{2})

Simplify

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

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

Remove parentheses:

(- a) = - a

= - 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

= - 3

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Frequently Asked Questions (FAQ)

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