What is the value of (sin(15)+sin(75))^6 ?

The value of (sin(15)+sin(75))^6 is 27/8

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Popular >

(sin(15)+sin(75))^6

Solution

(sin (1 5^{\circ}) + sin (7 5^{\circ}))^{6}

Solution

\frac{27}{8}

Decimal Notation

3.375

Solution steps

(sin (1 5^{\circ}) + sin (7 5^{\circ}))^{6}

Rewrite using trig identities:

sin (1 5^{\circ}) + sin (7 5^{\circ}) = 2 sin (4 5^{\circ}) cos (3 0^{\circ})

sin (1 5^{\circ}) + sin (7 5^{\circ})

Use the Sum to Product identity:

sin (s) + sin (t) = 2 sin (\frac{s + t}{2}) cos (\frac{s - t}{2})

= 2 sin (\frac{1 5 ^{\circ} + 7 5 ^{\circ}}{2}) cos (\frac{1 5 ^{\circ} - 7 5 ^{\circ}}{2})

Simplify

= 2 sin (4 5^{\circ}) cos (- 3 0^{\circ})

Use the following property:

cos (- x) = cos (x)

cos (- 3 0^{\circ}) = cos (3 0^{\circ})

= 2 sin (4 5^{\circ}) cos (3 0^{\circ})

= (2 sin (4 5^{\circ}) cos (3 0^{\circ}))^{6}

Simplify

= 64 sin^{6} (4 5^{\circ}) cos^{6} (3 0^{\circ})

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 (3 0^{\circ}) = \frac{3}{2}

cos (3 0^{\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{3}{2}

= 64 (\frac{2}{2})^{6} (\frac{3}{2})^{6}

Simplify

64 (\frac{2}{2})^{6} (\frac{3}{2})^{6} : \frac{27}{8}

64 (\frac{2}{2})^{6} (\frac{3}{2})^{6}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(\frac{2}{2})^{6} (\frac{3}{2})^{6} = (\frac{2}{2} \cdot \frac{3}{2})^{6}

= 64 (\frac{2}{2} \cdot \frac{3}{2})^{6}

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

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

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

\frac{2}{2}

Apply radical rule:

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

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

Apply exponent rule:

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

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

Apply radical rule:

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

= \frac{1}{2}

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

Multiply fractions:

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

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

Multiply:

1 \cdot 3 = 3

= \frac{3}{2 2}

= 64 (\frac{3}{2 2})^{6}

(\frac{3}{2 2})^{6} = \frac{3 ^{3}}{2 ^{9}}

(\frac{3}{2 2})^{6}

Apply exponent rule:

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

= \frac{( 3 ) ^{6}}{( 2 2 ) ^{6}}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(22)^{6} = 2^{6} (2)^{6}

= \frac{( 3 ) ^{6}}{2 ^{6} ( 2 ) ^{6}}

(3)^{6} : 3^{3}

Apply radical rule:

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

= (3^{\frac{1}{2}})^{6}

Apply exponent rule:

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

= 3^{\frac{1}{2} \cdot 6}

\frac{1}{2} \cdot 6 = 3

\frac{1}{2} \cdot 6

Multiply fractions:

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

= \frac{1 \cdot 6}{2}

Multiply the numbers:

1 \cdot 6 = 6

= \frac{6}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3

= 3^{3}

= \frac{3 ^{3}}{2 ^{6} ( 2 ) ^{6}}

(2)^{6} : 2^{3}

Apply radical rule:

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

= (2^{\frac{1}{2}})^{6}

Apply exponent rule:

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

= 2^{\frac{1}{2} \cdot 6}

\frac{1}{2} \cdot 6 = 3

\frac{1}{2} \cdot 6

Multiply fractions:

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

= \frac{1 \cdot 6}{2}

Multiply the numbers:

1 \cdot 6 = 6

= \frac{6}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3

= 2^{3}

= \frac{3 ^{3}}{2 ^{6} \cdot 2 ^{3}}

2^{6} \cdot 2^{3} = 2^{9}

2^{6} \cdot 2^{3}

Apply exponent rule:

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

2^{6} \cdot 2^{3} = 2^{6 + 3}

= 2^{6 + 3}

Add the numbers:

6 + 3 = 9

= 2^{9}

= \frac{3 ^{3}}{2 ^{9}}

= 64 \cdot \frac{3 ^{3}}{2 ^{9}}

Multiply fractions:

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

= \frac{3 ^{3} \cdot 64}{2 ^{9}}

Factor

64 : 2^{6}

Factor

64 = 2^{6}

= \frac{2 ^{6} \cdot 3 ^{3}}{2 ^{9}}

Cancel

\frac{3 ^{3} \cdot 2 ^{6}}{2 ^{9}} : \frac{3 ^{3}}{2 ^{3}}

\frac{3 ^{3} \cdot 2 ^{6}}{2 ^{9}}

Apply exponent rule:

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

\frac{2 ^{6}}{2 ^{9}} = \frac{1}{2 ^{9 - 6}}

= \frac{3 ^{3}}{2 ^{9 - 6}}

Subtract the numbers:

9 - 6 = 3

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

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

3^{3} = 27

= \frac{27}{2 ^{3}}

2^{3} = 8

= \frac{27}{8}

= \frac{27}{8}

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Frequently Asked Questions (FAQ)

What is the value of (sin(15)+sin(75))^6 ?
The value of (sin(15)+sin(75))^6 is 27/8