What is the value of sin^2((7pi)/6) ?

The value of sin^2((7pi)/6) is 1/4

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sin^2((7pi)/6)

Solution

sin^{2} (\frac{7 π}{6})

Solution

\frac{1}{4}

Decimal Notation

0.25

Solution steps

sin^{2} (\frac{7 π}{6})

Rewrite using trig identities:

\frac{1 - cos ( \frac{π}{3} )}{2}

sin^{2} (\frac{7 π}{6})

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{7 π}{6} )}{2}

Simplify:

2 \cdot \frac{7 π}{6} = \frac{7 π}{3}

2 \cdot \frac{7 π}{6}

Multiply fractions:

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

= \frac{7 π 2}{6}

Multiply the numbers:

7 \cdot 2 = 14

= \frac{14 π}{6}

Cancel the common factor:

2

= \frac{7 π}{3}

= \frac{1 - cos ( \frac{7 π}{3} )}{2}

cos (\frac{7 π}{3}) = cos (\frac{π}{3})

cos (\frac{7 π}{3})

Rewrite

\frac{7 π}{3}

2 π + \frac{π}{3}

= cos (2 π + \frac{π}{3})

Apply the periodicity of

cos

cos (x + 2 π) = cos (x)

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

= cos (\frac{π}{3})

= \frac{1 - cos ( \frac{π}{3} )}{2}

= \frac{1 - cos ( \frac{π}{3} )}{2}

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

cos (\frac{π}{3})

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{1}{2}

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

Simplify

\frac{1 - \frac{1}{2}}{2} : \frac{1}{4}

\frac{1 - \frac{1}{2}}{2}

Join

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

1 - \frac{1}{2}

Convert element to fraction:

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

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

1 \cdot 2 - 1 = 1

1 \cdot 2 - 1

Multiply the numbers:

1 \cdot 2 = 2

= 2 - 1

Subtract the numbers:

2 - 1 = 1

= 1

= \frac{1}{2}

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

= \frac{1}{4}

Popular Examples

csc((9pi)/8)

csc (\frac{9 π}{8})

tan((5pi)/6+(7pi)/4)

tan (\frac{5 π}{6} + \frac{7 π}{4})

2sec(pi/3)

2 sec (\frac{π}{3})

tan(2arccos(4/5))

tan (2 arccos (\frac{4}{5}))

sec(1020)

sec (102 0^{\circ})

Frequently Asked Questions (FAQ)

What is the value of sin^2((7pi)/6) ?
The value of sin^2((7pi)/6) is 1/4