What is the value of sin(105)sin(75) ?

The value of sin(105)sin(75) is (sqrt(3)+2)/4

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sin(105)sin(75)

sin (10 5^{\circ}) sin (7 5^{\circ})

\frac{3 + 2}{4}

Decimal Notation

0.93301 \dots

Solution steps

sin (10 5^{\circ}) sin (7 5^{\circ})

Rewrite using trig identities:

\frac{cos ( 3 0 ^{\circ} ) - cos ( 18 0 ^{\circ} )}{2}

sin (10 5^{\circ}) sin (7 5^{\circ})

Use the Product to Sum identity:

sin (s) sin (t) = \frac{1}{2} (cos (s - t) - cos (s + t))

= \frac{1}{2} (cos (10 5^{\circ} - 7 5^{\circ}) - cos (10 5^{\circ} + 7 5^{\circ}))

Simplify

= \frac{cos ( 3 0 ^{\circ} ) - cos ( 18 0 ^{\circ} )}{2}

= \frac{cos ( 3 0 ^{\circ} ) - cos ( 18 0 ^{\circ} )}{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}

Use the following trivial identity:

cos (18 0^{\circ}) = (- 1)

cos (18 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}

= (- 1)

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

Simplify

\frac{\frac{3}{2} - ( - 1 )}{2} : \frac{3 + 2}{4}

\frac{\frac{3}{2} - ( - 1 )}{2}

Apply rule

- (- a) = a

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

Join

\frac{3}{2} + 1 : \frac{3 + 2}{2}

\frac{3}{2} + 1

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{3 + 2}{2}

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 + 2}{4}

= \frac{3 + 2}{4}

250 sec^{2} (\frac{π}{3})

arccos (0.342)

tan (\frac{8}{7})

arccos (\frac{19}{406})

cos (\frac{10}{8})

What is the value of sin(105)sin(75) ?
The value of sin(105)sin(75) is (sqrt(3)+2)/4