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

The value of cos(2arccos(-1/3)) is -7/9

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

cos(2arccos(-1/3))

Solution

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

Solution

- \frac{7}{9}

Decimal Notation

- 0.77777 \dots

Solution steps

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

Rewrite using trig identities:

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

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

Use the Double Angle identity:

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

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

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

Rewrite using trig identities:

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

sin (arccos (- \frac{1}{3}))

Rewrite using trig identities:

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

Use the following identity:

sin (arccos (x)) = 1 - x^{2}

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

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

Simplify

= \frac{2 2}{3}

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

Simplify

1 - 2 (\frac{2 2}{3})^{2} : - \frac{7}{9}

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

2 (\frac{2 2}{3})^{2} = \frac{16}{9}

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

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

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

Apply exponent rule:

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

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

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(22)^{2} = 2^{2} (2)^{2}

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

(2)^{2} : 2

Apply radical rule:

a = a^{\frac{1}{2}}

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

Apply exponent rule:

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

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2

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

Multiply fractions:

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

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

2^{3} \cdot 2 = 2^{4}

2^{3} \cdot 2

Apply exponent rule:

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

2^{3} \cdot 2 = 2^{3 + 1}

= 2^{3 + 1}

Add the numbers:

3 + 1 = 4

= 2^{4}

= \frac{2 ^{4}}{3 ^{2}}

2^{4} = 16

= \frac{16}{3 ^{2}}

3^{2} = 9

= \frac{16}{9}

= 1 - \frac{16}{9}

Convert element to fraction:

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

= \frac{1 \cdot 9}{9} - \frac{16}{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 - 16}{9}

1 \cdot 9 - 16 = - 7

1 \cdot 9 - 16

Multiply the numbers:

1 \cdot 9 = 9

= 9 - 16

Subtract the numbers:

9 - 16 = - 7

= - 7

= \frac{- 7}{9}

Apply the fraction rule:

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

= - \frac{7}{9}

= - \frac{7}{9}

Popular Examples

cosh(45)

cosh (45)

-3cos(1)

- 3 cos (1)

sqrt(sin(1))

sin (1)

csc(61)

csc (6 1^{\circ})

arccos(-(sqrt(5))/5)

arccos (- \frac{5}{5})

Frequently Asked Questions (FAQ)

What is the value of cos(2arccos(-1/3)) ?
The value of cos(2arccos(-1/3)) is -7/9