What is the value of sin(105)sin(15) ?

The value of sin(105)sin(15) is 1/4

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sin(105)sin(15)

sin (10 5^{\circ}) sin (1 5^{\circ})

\frac{1}{4}

Decimal Notation

0.25

Solution steps

sin (10 5^{\circ}) sin (1 5^{\circ})

Rewrite using trig identities:

\frac{cos ( 9 0 ^{\circ} ) - cos ( 12 0 ^{\circ} )}{2}

sin (10 5^{\circ}) sin (1 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} - 1 5^{\circ}) - cos (10 5^{\circ} + 1 5^{\circ}))

Simplify

= \frac{cos ( 9 0 ^{\circ} ) - cos ( 12 0 ^{\circ} )}{2}

= \frac{cos ( 9 0 ^{\circ} ) - cos ( 12 0 ^{\circ} )}{2}

Use the following trivial identity:

cos (9 0^{\circ}) = 0

cos (9 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}

= 0

Use the following trivial identity:

cos (12 0^{\circ}) = - \frac{1}{2}

cos (12 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{1}{2}

= \frac{0 - ( - \frac{1}{2} )}{2}

Simplify

\frac{0 - ( - \frac{1}{2} )}{2} : \frac{1}{4}

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

Apply rule

- (- a) = a

= \frac{0 + \frac{1}{2}}{2}

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

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

= \frac{1}{4}

sec^{2} (- π)

cos (180 0^{\circ})

2 csc^{2} (\frac{π}{3}) cot (\frac{π}{3})

30 sin (4 0^{\circ})

sec (\frac{5}{4})