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

The value of cos(2arcsin(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(2arcsin(1/3))

cos (2 arcsin (\frac{1}{3}))

\frac{7}{9}

Decimal Notation

0.77777 \dots

Solution steps

cos (2 arcsin (\frac{1}{3}))

Rewrite using trig identities:

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

cos (2 arcsin (\frac{1}{3}))

Use the Double Angle identity:

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

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

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

Rewrite using trig identities:

sin (arcsin (\frac{1}{3})) = \frac{1}{3}

Use the following identity:

sin (arcsin (x)) = x

= \frac{1}{3}

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

Simplify

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

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

2 (\frac{1}{3})^{2} = \frac{2}{9}

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

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

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

Apply exponent rule:

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

= \frac{1 ^{2}}{3 ^{2}}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{3 ^{2}}

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

3^{2} = 9

= \frac{2}{9}

= 1 - \frac{2}{9}

Convert element to fraction:

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

= \frac{1 \cdot 9}{9} - \frac{2}{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 - 2}{9}

1 \cdot 9 - 2 = 7

1 \cdot 9 - 2

Multiply the numbers:

1 \cdot 9 = 9

= 9 - 2

Subtract the numbers:

9 - 2 = 7

= 7

= \frac{7}{9}

= \frac{7}{9}

\frac{sin ( \frac{π}{4} )}{cos ( \frac{π}{4} )}

arccot (\frac{4}{3})

tan (\frac{5 π}{6} + \frac{5 π}{4})

cos (12 0^{\circ} - 4 5^{\circ})

arccos (0.866)