What is the value of cos(2arccos(-2/3)) ?

The value of cos(2arccos(-2/3)) is -1/9

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cos(2arccos(-2/3))

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

- \frac{1}{9}

Decimal Notation

- 0.11111 \dots

Solution steps

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

Rewrite using trig identities:

2 cos^{2} (arccos (- \frac{2}{3})) - 1

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

Use the Double Angle identity:

cos (2 x) = 2 cos^{2} (x) - 1

= 2 cos^{2} (arccos (- \frac{2}{3})) - 1

= 2 cos^{2} (arccos (- \frac{2}{3})) - 1

Rewrite using trig identities:

cos (arccos (- \frac{2}{3})) = - \frac{2}{3}

Use the following identity:

cos (arccos (x)) = x

= - \frac{2}{3}

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

Simplify

2 (- \frac{2}{3})^{2} - 1 : - \frac{1}{9}

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

2 (- \frac{2}{3})^{2} = \frac{8}{9}

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

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

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

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

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

= (\frac{2}{3})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{2 ^{2}}{3 ^{2}}

= 2 \cdot \frac{2 ^{2}}{3 ^{2}}

Multiply fractions:

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

= \frac{2 ^{2} \cdot 2}{3 ^{2}}

2^{2} \cdot 2 = 2^{3}

2^{2} \cdot 2

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

2^{2} \cdot 2 = 2^{2 + 1}

= 2^{2 + 1}

Add the numbers:

2 + 1 = 3

= 2^{3}

= \frac{2 ^{3}}{3 ^{2}}

2^{3} = 8

= \frac{8}{3 ^{2}}

3^{2} = 9

= \frac{8}{9}

= \frac{8}{9} - 1

Convert element to fraction:

1 = \frac{1 \cdot 9}{9}

= - \frac{1 \cdot 9}{9} + \frac{8}{9}

Since the denominators are equal, combine the fractions:

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

= \frac{- 1 \cdot 9 + 8}{9}

- 1 \cdot 9 + 8 = - 1

- 1 \cdot 9 + 8

Multiply the numbers:

1 \cdot 9 = 9

= - 9 + 8

Add/Subtract the numbers:

- 9 + 8 = - 1

= - 1

= \frac{- 1}{9}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{9}

= - \frac{1}{9}

cos (\frac{π}{4}) - cos (\frac{3 π}{4})

sec (15 π)

3 cos (5 3^{\circ})

1 - \frac{1}{3} sin^{2} (\frac{π}{3})

\frac{7 sin ( - 0.001 )}{- 0.001}

What is the value of cos(2arccos(-2/3)) ?
The value of cos(2arccos(-2/3)) is -1/9