What is the value of sin^2(1/2 arccos(4/5)) ?

The value of sin^2(1/2 arccos(4/5)) is 1/10

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sin^2(1/2 arccos(4/5))

Solution

sin^{2} (\frac{1}{2} arccos (\frac{4}{5}))

Solution

\frac{1}{10}

Decimal Notation

0.1

Solution steps

sin^{2} (\frac{1}{2} arccos (\frac{4}{5}))

Simplify:

\frac{1}{2} arccos (\frac{4}{5}) = \frac{arccos ( \frac{4}{5} )}{2}

\frac{1}{2} arccos (\frac{4}{5})

Multiply fractions:

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

= \frac{1 \cdot arccos ( \frac{4}{5} )}{2}

Multiply:

1 \cdot arccos (\frac{4}{5}) = arccos (\frac{4}{5})

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

= sin^{2} (\frac{arccos ( \frac{4}{5} )}{2})

Rewrite using trig identities:

\frac{1 - cos ( arccos ( \frac{4}{5} ) )}{2}

sin^{2} (\frac{arccos ( \frac{4}{5} )}{2})

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{a r c c o s ( \frac{4}{5} )}{2} )}{2}

Simplify:

2 \cdot \frac{arccos ( \frac{4}{5} )}{2} = arccos (\frac{4}{5})

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

Multiply fractions:

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

= \frac{arccos ( \frac{4}{5} ) \cdot 2}{2}

Cancel the common factor:

2

= arccos (\frac{4}{5})

= \frac{1 - cos ( arccos ( \frac{4}{5} ) )}{2}

= \frac{1 - cos ( arccos ( \frac{4}{5} ) )}{2}

Rewrite using trig identities:

cos (arccos (\frac{4}{5})) = \frac{4}{5}

Use the following identity:

cos (arccos (x)) = x

= \frac{4}{5}

= \frac{1 - \frac{4}{5}}{2}

Simplify

\frac{1 - \frac{4}{5}}{2} : \frac{1}{10}

\frac{1 - \frac{4}{5}}{2}

Join

1 - \frac{4}{5} : \frac{1}{5}

1 - \frac{4}{5}

Convert element to fraction:

1 = \frac{1 \cdot 5}{5}

= \frac{1 \cdot 5}{5} - \frac{4}{5}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 5 - 4}{5}

1 \cdot 5 - 4 = 1

1 \cdot 5 - 4

Multiply the numbers:

1 \cdot 5 = 5

= 5 - 4

Subtract the numbers:

5 - 4 = 1

= 1

= \frac{1}{5}

= \frac{\frac{1}{5}}{2}

Apply the fraction rule:

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

= \frac{1}{5 \cdot 2}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{1}{10}

= \frac{1}{10}

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Frequently Asked Questions (FAQ)

What is the value of sin^2(1/2 arccos(4/5)) ?
The value of sin^2(1/2 arccos(4/5)) is 1/10