What is the value of (sin^2(45)cos^2(45))^{3/2} ?

The value of (sin^2(45)cos^2(45))^{3/2} is 1/8

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(sin^2(45)cos^2(45))^{3/2}

Solution

(sin^{2} (4 5^{\circ}) cos^{2} (4 5^{\circ}))^{\frac{3}{2}}

Solution

\frac{1}{8}

Decimal Notation

0.125

Solution steps

(sin^{2} (4 5^{\circ}) cos^{2} (4 5^{\circ}))^{\frac{3}{2}}

Use the following trivial identity:

sin (4 5^{\circ}) = \frac{2}{2}

sin (4 5^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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}

Use the following trivial identity:

cos (4 5^{\circ}) = \frac{2}{2}

cos (4 5^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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}

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

Simplify

(\frac{2}{2})^{2} (\frac{2}{2})^{2}^{\frac{3}{2}} : \frac{1}{8}

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

Simplify

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

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

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

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

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

Add the numbers:

2 + 2 = 4

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

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

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

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

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

\frac{2}{2}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

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

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

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

Apply exponent rule:

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

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

(2^{\frac{1}{2}})^{4} : 2^{2}

Apply exponent rule:

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

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

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

\frac{1}{2} \cdot 4

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

= 2^{2}

= \frac{1 ^{4}}{2 ^{2}}

Apply rule

1^{a} = 1

1^{4} = 1

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

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

Apply exponent rule:

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

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

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

Apply exponent rule:

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

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

2 \cdot \frac{3}{2} = 3

2 \cdot \frac{3}{2}

Multiply fractions:

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

= \frac{3 \cdot 2}{2}

Cancel the common factor:

2

= 3

= 2^{3}

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

1^{\frac{3}{2}} = 1 \cdot 1^{\frac{1}{2}}

1^{\frac{3}{2}}

1^{\frac{3}{2}} = 1^{1 + \frac{1}{2}}

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

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

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

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

Multiply:

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

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

Apply rule

1^{a} = 1

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

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

2^{3} = 8

= \frac{1}{8}

= \frac{1}{8}

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Frequently Asked Questions (FAQ)

What is the value of (sin^2(45)cos^2(45))^{3/2} ?
The value of (sin^2(45)cos^2(45))^{3/2} is 1/8