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

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

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

Solution

sin^{2} (\frac{5 π}{12})

Solution

\frac{2 + 3}{4}

Decimal Notation

0.93301 \dots

Solution steps

sin^{2} (\frac{5 π}{12})

Rewrite using trig identities:

\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}

= \frac{1 - cos ( \frac{5 π}{6} )}{2}

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}

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

Simplify

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

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

Apply rule

- (- a) = a

= \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}

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Frequently Asked Questions (FAQ)

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