What is the value of cos((2pi)/3)-cos((4pi)/3) ?

The value of cos((2pi)/3)-cos((4pi)/3) is 0

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

cos((2pi)/3)-cos((4pi)/3)

cos (\frac{2 π}{3}) - cos (\frac{4 π}{3})

0

Solution steps

cos (\frac{2 π}{3}) - cos (\frac{4 π}{3})

Rewrite using trig identities:

cos (\frac{4 π}{3}) = 2 cos^{2} (\frac{2 π}{3}) - 1

cos (\frac{4 π}{3})

Write

cos (\frac{4 π}{3})

cos (2 \cdot \frac{2 π}{3})

= cos (2 \cdot \frac{2 π}{3})

Use the Double Angle identity:

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

= 2 cos^{2} (\frac{2 π}{3}) - 1

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

Simplify

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

Use the following trivial identity:

cos (\frac{2 π}{3}) = - \frac{1}{2}

cos (\frac{2 π}{3})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - \frac{1}{2}

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

Simplify

- \frac{1}{2} - 2 (- \frac{1}{2})^{2} + 1 : 0

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

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

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

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

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

Apply exponent rule:

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

n

is even

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

Cancel the common factor:

2

= \frac{1}{2}

= - \frac{1}{2} - \frac{1}{2} + 1

Combine the fractions

- \frac{1}{2} - \frac{1}{2} : - 1

Apply rule

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

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

Subtract the numbers:

- 1 - 1 = - 2

= \frac{- 2}{2}

Apply the fraction rule:

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

= - \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= - 1

= - 1 + 1

Add/Subtract the numbers:

- 1 + 1 = 0

= 0

= 0

What is the value of cos((2pi)/3)-cos((4pi)/3) ?
The value of cos((2pi)/3)-cos((4pi)/3) is 0