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

The value of sin(2arccos(2/3)) is (4sqrt(5))/9

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

sin(2arccos(2/3))

sin (2 arccos (\frac{2}{3}))

\frac{4 5}{9}

Decimal

0.99380 \dots

Solution steps

sin (2 arccos (\frac{2}{3}))

Rewrite using trig identities:

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

sin (2 arccos (\frac{2}{3}))

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

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

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

Rewrite using trig identities:

sin (arccos (\frac{2}{3})) = \frac{5}{3}

sin (arccos (\frac{2}{3}))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{5}{3}

Rewrite using trig identities:

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

Use the following identity:

cos (arccos (x)) = x

= \frac{2}{3}

= 2 \cdot \frac{5}{3} \cdot \frac{2}{3}

Simplify

2 \cdot \frac{5}{3} \cdot \frac{2}{3} : \frac{4 5}{9}

2 \cdot \frac{5}{3} \cdot \frac{2}{3}

Multiply fractions:

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

= \frac{5 \cdot 2 \cdot 2}{3 \cdot 3}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 5}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{4 5}{9}

= \frac{4 5}{9}

sin^{2} (\frac{2 π}{3}) + cos^{2} (\frac{2 π}{3})

tan (arcsin (\frac{- 3}{2}))

2 sin (3 0^{\circ}) + 4 cos (6 0^{\circ}) - 3 cot (4 5^{\circ}) + sec (4 5^{\circ})

5 + 4 cos (\frac{π}{6})

arcsin (cos (- 6 0^{\circ}))

What is the value of sin(2arccos(2/3)) ?
The value of sin(2arccos(2/3)) is (4sqrt(5))/9