What is the value of cos^2(1/2 arccos(12/13)) ?

The value of cos^2(1/2 arccos(12/13)) is 25/26

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cos^2(1/2 arccos(12/13))

Solution

cos^{2} (\frac{1}{2} arccos (\frac{12}{13}))

Solution

\frac{25}{26}

Decimal Notation

0.96153 \dots

Solution steps

cos^{2} (\frac{1}{2} arccos (\frac{12}{13}))

Simplify:

\frac{1}{2} arccos (\frac{12}{13}) = \frac{arccos ( \frac{12}{13} )}{2}

\frac{1}{2} arccos (\frac{12}{13})

Multiply fractions:

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

= \frac{1 \cdot arccos ( \frac{12}{13} )}{2}

Multiply:

1 \cdot arccos (\frac{12}{13}) = arccos (\frac{12}{13})

= \frac{arccos ( \frac{12}{13} )}{2}

= cos^{2} (\frac{arccos ( \frac{12}{13} )}{2})

Rewrite using trig identities:

\frac{1 + cos ( arccos ( \frac{12}{13} ) )}{2}

cos^{2} (\frac{arccos ( \frac{12}{13} )}{2})

Use the following identity:

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

Use the Double Angle identity

cos (2 θ) = 2 cos^{2} (θ) - 1

Switch sides

2 cos^{2} (θ) - 1 = cos (2 θ)

Add

1

to both sides

2 sin^{2} (θ) = 1 + cos (2 θ)

Divide both sides by

2

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

= \frac{1 + cos ( 2 \cdot \frac{a r c c o s ( \frac{12}{13} )}{2} )}{2}

Simplify:

2 \cdot \frac{arccos ( \frac{12}{13} )}{2} = arccos (\frac{12}{13})

2 \cdot \frac{arccos ( \frac{12}{13} )}{2}

Multiply fractions:

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

= \frac{arccos ( \frac{12}{13} ) \cdot 2}{2}

Cancel the common factor:

2

= arccos (\frac{12}{13})

= \frac{1 + cos ( arccos ( \frac{12}{13} ) )}{2}

= \frac{1 + cos ( arccos ( \frac{12}{13} ) )}{2}

Rewrite using trig identities:

cos (arccos (\frac{12}{13})) = \frac{12}{13}

Use the following identity:

cos (arccos (x)) = x

= \frac{12}{13}

= \frac{1 + \frac{12}{13}}{2}

Simplify

\frac{1 + \frac{12}{13}}{2} : \frac{25}{26}

\frac{1 + \frac{12}{13}}{2}

Join

1 + \frac{12}{13} : \frac{25}{13}

1 + \frac{12}{13}

Convert element to fraction:

1 = \frac{1 \cdot 13}{13}

= \frac{1 \cdot 13}{13} + \frac{12}{13}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 13 + 12}{13}

1 \cdot 13 + 12 = 25

1 \cdot 13 + 12

Multiply the numbers:

1 \cdot 13 = 13

= 13 + 12

Add the numbers:

13 + 12 = 25

= 25

= \frac{25}{13}

= \frac{\frac{25}{13}}{2}

Apply the fraction rule:

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

= \frac{25}{13 \cdot 2}

Multiply the numbers:

13 \cdot 2 = 26

= \frac{25}{26}

= \frac{25}{26}

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Frequently Asked Questions (FAQ)

What is the value of cos^2(1/2 arccos(12/13)) ?
The value of cos^2(1/2 arccos(12/13)) is 25/26