What is the value of 2sin(37.5)sin(97.5) ?

The value of 2sin(37.5)sin(97.5) is (1+sqrt(2))/2

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2sin(37.5)sin(97.5)

2 sin (37. 5^{\circ}) sin (97. 5^{\circ})

\frac{1 + 2}{2}

Decimal Notation

1.20710 \dots

Solution steps

2 sin (37. 5^{\circ}) sin (97. 5^{\circ})

Rewrite using trig identities:

cos (6 0^{\circ}) - cos (13 5^{\circ})

2 sin (37. 5^{\circ}) sin (97. 5^{\circ})

Use the Product to Sum identity:

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

= 2 \cdot \frac{1}{2} (cos (37. 5^{\circ} - 97. 5^{\circ}) - cos (37. 5^{\circ} + 97. 5^{\circ}))

Simplify

= cos (- 6 0^{\circ}) - cos (13 5^{\circ})

Use the following property:

cos (- x) = cos (x)

cos (- 6 0^{\circ}) = cos (6 0^{\circ})

= cos (6 0^{\circ}) - cos (13 5^{\circ})

= cos (6 0^{\circ}) - cos (13 5^{\circ})

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}

Use the following trivial identity:

cos (13 5^{\circ}) = - \frac{2}{2}

cos (13 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}

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

Simplify

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

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

Apply rule

- (- a) = a

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

Apply rule

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

= \frac{1 + 2}{2}

= \frac{1 + 2}{2}

arcsin (sin (\frac{π}{7}))

2 cos (0) + sin (2 \cdot 0)

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

- arcsin (1)

2 sin (\frac{π}{3}) cos (\frac{π}{3})

What is the value of 2sin(37.5)sin(97.5) ?
The value of 2sin(37.5)sin(97.5) is (1+sqrt(2))/2