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

The value of cos(105)sin(15) is (sqrt(3)-2)/4

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cos(105)sin(15)

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

\frac{3 - 2}{4}

Decimal Notation

- 0.06698 \dots

Solution steps

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

Rewrite using trig identities:

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

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

Use the Product to Sum identity:

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

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

Simplify

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

Use the following property:

sin (- x) = - sin (x)

sin (- 9 0^{\circ}) = - sin (9 0^{\circ})

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

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

Use the following trivial identity:

sin (12 0^{\circ}) = \frac{3}{2}

sin (12 0^{\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{3}{2}

Use the following trivial identity:

sin (9 0^{\circ}) = 1

sin (9 0^{\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}

= 1

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

Simplify

\frac{\frac{3}{2} - 1}{2} : \frac{3 - 2}{4}

\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}

40 \cdot cos (6 0^{\circ})

5 sin (\frac{π}{6})

arccos (2 π)

arccos (cos (\frac{6 π}{5}))

cos (- 0.3)

What is the value of cos(105)sin(15) ?
The value of cos(105)sin(15) is (sqrt(3)-2)/4