Is cos((3pi)/2*123)=cos(pi/2*123) ?

The answer to whether cos((3pi)/2*123)=cos(pi/2*123) is True

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prove cos((3pi)/2123)=cos(pi/2123)

Solution

prove

cos (\frac{3 π}{2} \cdot 123) = cos (\frac{π}{2} \cdot 123)

Solution

True

Solution steps

cos (\frac{3 π}{2} \cdot 123) = cos (\frac{π}{2} \cdot 123)

Manipulating left side

cos (\frac{3 π}{2} 123)

Simplify

cos (123 \frac{3 π}{2}) : 0

cos (123 \cdot \frac{3 π}{2})

Multiply

123 \cdot \frac{3 π}{2} : \frac{369 π}{2}

123 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 123}{2}

Multiply the numbers:

3 \cdot 123 = 369

= \frac{369 π}{2}

= cos (\frac{369 π}{2})

cos (\frac{369 π}{2}) = cos (\frac{π}{2})

cos (\frac{369 π}{2})

Rewrite

\frac{369 π}{2}

2 π \cdot 92 + \frac{π}{2}

= cos (2 π 92 + \frac{π}{2})

Apply the periodicity of

cos

cos (x + 2 π \cdot k) = cos (x)

cos (2 π \cdot 92 + \frac{π}{2}) = cos (\frac{π}{2})

= cos (\frac{π}{2})

= cos (\frac{π}{2})

Use the following trivial identity:

cos (\frac{π}{2}) = 0

cos (\frac{π}{2})

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}

= 0

= 0

= 0

Manipulating right side

cos (\frac{π}{2} 123)

Simplify

cos (123 \frac{π}{2}) : 0

cos (123 \cdot \frac{π}{2})

Multiply

123 \cdot \frac{π}{2} : \frac{123 π}{2}

123 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 123}{2}

= cos (\frac{π 123}{2})

Rewrite using trig identities:

1 - 2 sin^{2} (\frac{3 π}{4})

cos (\frac{123 π}{2})

Write

cos (\frac{123 π}{2})

cos (2 \cdot \frac{123 π}{4})

= cos (2 \cdot \frac{123 π}{4})

Use the Double Angle identity:

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

= 1 - 2 sin^{2} (\frac{123 π}{4})

sin (\frac{123 π}{4}) = sin (\frac{3 π}{4})

sin (\frac{123 π}{4})

Rewrite

\frac{123 π}{4}

2 π \cdot 15 + \frac{3 π}{4}

= sin (2 π 15 + \frac{3 π}{4})

Apply the periodicity of

sin

sin (x + 2 π \cdot k) = sin (x)

sin (2 π \cdot 15 + \frac{3 π}{4}) = sin (\frac{3 π}{4})

= sin (\frac{3 π}{4})

= 1 - 2 sin^{2} (\frac{3 π}{4})

= 1 - 2 sin^{2} (\frac{3 π}{4})

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}

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

Simplify

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

1 - 2 (\frac{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

= 1 - 1

Subtract the numbers:

1 - 1 = 0

= 0

= 0

= 0

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is cos((3pi)/2*123)=cos(pi/2*123) ?
The answer to whether cos((3pi)/2*123)=cos(pi/2*123) is True

prove cos((3pi)/2*123)=cos(pi/2*123)

Solution

Solution

Popular Examples

Frequently Asked Questions (FAQ)

Is cos((3pi)/2*123)=cos(pi/2*123) ?

prove cos((3pi)/2123)=cos(pi/2123)

Is cos((3pi)/2123)=cos(pi/2123) ?