What is the value of 8cos(22.5)*cos(67.5) ?

The value of 8cos(22.5)*cos(67.5) is 2sqrt(2)

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8cos(22.5)*cos(67.5)

8 cos (22. 5^{\circ}) \cdot cos (67. 5^{\circ})

22

Decimal Notation

2.82842 \dots

Solution steps

8 cos (22. 5^{\circ}) cos (67. 5^{\circ})

Rewrite using trig identities:

4 (cos (9 0^{\circ}) + cos (4 5^{\circ}))

8 cos (22. 5^{\circ}) cos (67. 5^{\circ})

Use the Product to Sum identity:

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

= 8 \cdot \frac{1}{2} (cos (22. 5^{\circ} - 67. 5^{\circ}) + cos (22. 5^{\circ} + 67. 5^{\circ}))

Simplify

= 4 (cos (9 0^{\circ}) + cos (- 4 5^{\circ}))

Use the following property:

cos (- x) = cos (x)

cos (- 4 5^{\circ}) = cos (4 5^{\circ})

= 4 (cos (9 0^{\circ}) + cos (4 5^{\circ}))

= 4 (cos (9 0^{\circ}) + cos (4 5^{\circ}))

Use the following trivial identity:

cos (9 0^{\circ}) = 0

cos (9 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}

= 0

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}

= 4 (0 + \frac{2}{2})

Simplify

4 (0 + \frac{2}{2}) : 22

4 (0 + \frac{2}{2})

0 + \frac{2}{2} = \frac{2}{2}

= 4 \cdot \frac{2}{2}

Multiply fractions:

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

= \frac{2 \cdot 4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 22

= 22

\frac{8}{tan ( 6 0 ^{\circ} )}

cot (arctan (\frac{3}{4}))

3 (cos (\frac{π}{2}) + i sin (\frac{π}{2}))

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

3 tan (2 0^{\circ})

What is the value of 8cos(22.5)*cos(67.5) ?
The value of 8cos(22.5)*cos(67.5) is 2sqrt(2)