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

The value of -3+4cos((4pi)/3) is -5

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

Solution

- 3 + 4 cos (\frac{4 π}{3})

Solution

- 5

Solution steps

- 3 + 4 cos (\frac{4 π}{3})

Rewrite using trig identities:

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

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

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

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

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}

Use the following trivial identity:

sin (\frac{2 π}{3}) = \frac{3}{2}

sin (\frac{2 π}{3})

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

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

Simplify

- 3 + 4 (- \frac{1}{2})^{2} - (\frac{3}{2})^{2} : - 5

- 3 + 4 (- \frac{1}{2})^{2} - (\frac{3}{2})^{2}

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

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

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

(- \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^{2} = 4

= \frac{1}{4}

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

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

Apply exponent rule:

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

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

(3)^{2} : 3

Apply radical rule:

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

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

Apply exponent rule:

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

= 3^{\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

= 3

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

2^{2} = 4

= \frac{3}{4}

= 4 (\frac{1}{4} - \frac{3}{4})

Simplify

\frac{1}{4} - \frac{3}{4} : - \frac{1}{2}

\frac{1}{4} - \frac{3}{4}

Apply rule

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

= \frac{1 - 3}{4}

Subtract the numbers:

1 - 3 = - 2

= \frac{- 2}{4}

Apply the fraction rule:

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

= - \frac{2}{4}

Cancel the common factor:

2

= - \frac{1}{2}

= 4 (- \frac{1}{2})

Remove parentheses:

(- a) = - a

= - 4 \cdot \frac{1}{2}

Multiply fractions:

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

= - \frac{1 \cdot 4}{2}

Multiply the numbers:

1 \cdot 4 = 4

= - \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= - 2

= - 3 - 2

Subtract the numbers:

- 3 - 2 = - 5

= - 5

= - 5

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Frequently Asked Questions (FAQ)

What is the value of -3+4cos((4pi)/3) ?
The value of -3+4cos((4pi)/3) is -5