What is the value of cos(37.5)sin(7.5) ?

The value of cos(37.5)sin(7.5) is (sqrt(2)-1)/4

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cos(37.5)sin(7.5)

cos (37. 5^{\circ}) sin (7. 5^{\circ})

\frac{2 - 1}{4}

Decimal Notation

0.10355 \dots

Solution steps

cos (37. 5^{\circ}) sin (7. 5^{\circ})

Rewrite using trig identities:

\frac{sin ( 4 5 ^{\circ} ) - sin ( 3 0 ^{\circ} )}{2}

cos (37. 5^{\circ}) sin (7. 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 (7. 5^{\circ} + 37. 5^{\circ}) + sin (7. 5^{\circ} - 37. 5^{\circ}))

Simplify

= \frac{sin ( 4 5 ^{\circ} ) + sin ( - 3 0 ^{\circ} )}{2}

Use the following property:

sin (- x) = - sin (x)

sin (- 3 0^{\circ}) = - sin (3 0^{\circ})

= \frac{sin ( 4 5 ^{\circ} ) - sin ( 3 0 ^{\circ} )}{2}

= \frac{sin ( 4 5 ^{\circ} ) - sin ( 3 0 ^{\circ} )}{2}

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:

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 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{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 (6 6^{\circ})

1 - cos (2 π)

2 tan (\frac{π}{4})

sec (- 2 π)

7 cos (3 0^{\circ})

What is the value of cos(37.5)sin(7.5) ?
The value of cos(37.5)sin(7.5) is (sqrt(2)-1)/4