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

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

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

Solution

2 - 4 cos (\frac{3 π}{2})

Solution

2

Solution steps

2 - 4 cos (\frac{3 π}{2})

Rewrite using trig identities:

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

cos (\frac{3 π}{2})

Write

cos (\frac{3 π}{2})

cos (2 \cdot \frac{3 π}{4})

= cos (2 \cdot \frac{3 π}{4})

Use the Double Angle identity:

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

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

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

Use the following trivial identity:

cos (\frac{3 π}{4}) = - \frac{2}{2}

cos (\frac{3 π}{4})

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{2}{2}

Use the following trivial identity:

sin (\frac{3 π}{4}) = \frac{2}{2}

sin (\frac{3 π}{4})

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{2}{2}

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

Simplify

2 - 4 (- \frac{2}{2})^{2} - (\frac{2}{2})^{2} : 2

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

4 (- \frac{2}{2})^{2} - (\frac{2}{2})^{2} = 0

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

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

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

Apply exponent rule:

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

n

is even

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

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

Apply exponent rule:

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

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

Cancel the common factor:

2

= \frac{1}{2}

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

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

Apply exponent rule:

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

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

Cancel the common factor:

2

= \frac{1}{2}

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

Add similar elements:

\frac{1}{2} - \frac{1}{2} = 0

= 4 \cdot 0

Apply rule

0 \cdot a = 0

= 0

= 2 - 0

Subtract the numbers:

2 - 0 = 2

= 2

= 2

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Frequently Asked Questions (FAQ)

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