What is the value of 5/2 cos((3pi)/2) ?

The value of 5/2 cos((3pi)/2) is 0

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5/2 cos((3pi)/2)

\frac{5}{2} cos (\frac{3 π}{2})

0

Solution steps

\frac{5}{2} cos (\frac{3 π}{2})

\frac{5}{2} cos (\frac{3 π}{2}) = \frac{5 cos ( \frac{3 π}{2} )}{2}

\frac{5}{2} cos (\frac{3 π}{2})

Multiply fractions:

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

= \frac{5 cos ( \frac{3 π}{2} )}{2}

= \frac{5 cos ( \frac{3 π}{2} )}{2}

Rewrite using trig identities:

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

= 1 - 2 sin^{2} (\frac{3 π}{4})

= \frac{5 ( 1 - 2 sin ^{2} ( \frac{3 π}{4} ) )}{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}

= \frac{5 ( 1 - 2 ( \frac{2}{2} ) ^{2} )}{2}

Simplify

\frac{5 ( 1 - 2 ( \frac{2}{2} ) ^{2} )}{2} : 0

\frac{5 ( 1 - 2 ( \frac{2}{2} ) ^{2} )}{2}

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= \frac{5 ( 1 - 1 )}{2}

5 (1 - 1) = 0

5 (1 - 1)

Subtract the numbers:

1 - 1 = 0

= 5 \cdot 0

Apply rule

0 \cdot a = 0

= 0

= \frac{0}{2}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

= 0

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