What is the value of sin(52.5)sin(7.5) ?

The value of sin(52.5)sin(7.5) is (sqrt(2)-1)/4

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sin(52.5)sin(7.5)

sin (52. 5^{\circ}) sin (7. 5^{\circ})

\frac{2 - 1}{4}

Decimal Notation

0.10355 \dots

Solution steps

sin (52. 5^{\circ}) sin (7. 5^{\circ})

Rewrite using trig identities:

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

sin (52. 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 (52. 5^{\circ} - 7. 5^{\circ}) - cos (52. 5^{\circ} + 7. 5^{\circ}))

Simplify

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

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

Use the following trivial identity:

cos (4 5^{\circ}) = \frac{2}{2}

cos (4 5^{\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{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}

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

Simplify

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

\frac{\frac{2}{2} - \frac{1}{2}}{2}

Combine the fractions

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

Apply rule

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

= \frac{2 - 1}{2}

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 - 1}{4}

= \frac{2 - 1}{4}

sin (arctan (- \frac{4}{3}))

2 cos (\frac{3 π}{4})

cos^{2} (1 8^{\circ})

3 tan (3 0^{\circ})

arcsin (sin (- \frac{5 π}{3}))

What is the value of sin(52.5)sin(7.5) ?
The value of sin(52.5)sin(7.5) is (sqrt(2)-1)/4