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

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

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sin(105)-sin(15)

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

\frac{2}{2}

Decimal Notation

0.70710 \dots

Solution steps

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

Rewrite using trig identities:

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

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

Simplify

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

= 2 sin (4 5^{\circ}) cos (6 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 (6 0^{\circ}) = \frac{1}{2}

cos (6 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}

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

Simplify

2 \cdot \frac{2}{2} \cdot \frac{1}{2} : \frac{2}{2}

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

Multiply fractions:

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

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

Cancel the common factor:

2

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

Multiply:

2 \cdot 1 = 2

= \frac{2}{2}

= \frac{2}{2}

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

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

arctan (\frac{- 1}{1})

csc (- 42 0^{\circ})

arccos (π)

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