What is the value of cos(105)cos(75) ?

The value of cos(105)cos(75) is (sqrt(3)-2)/4

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cos(105)cos(75)

cos (10 5^{\circ}) cos (7 5^{\circ})

\frac{3 - 2}{4}

Decimal Notation

- 0.06698 \dots

Solution steps

cos (10 5^{\circ}) cos (7 5^{\circ})

Rewrite using trig identities:

\frac{cos ( 3 0 ^{\circ} ) + cos ( 18 0 ^{\circ} )}{2}

cos (10 5^{\circ}) cos (7 5^{\circ})

Use the Product to Sum identity:

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

= \frac{1}{2} (cos (10 5^{\circ} - 7 5^{\circ}) + cos (10 5^{\circ} + 7 5^{\circ}))

Simplify

= \frac{cos ( 3 0 ^{\circ} ) + cos ( 18 0 ^{\circ} )}{2}

= \frac{cos ( 3 0 ^{\circ} ) + cos ( 18 0 ^{\circ} )}{2}

Use the following trivial identity:

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

cos (3 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{3}{2}

Use the following trivial identity:

cos (18 0^{\circ}) = (- 1)

cos (18 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}

= (- 1)

= \frac{\frac{3}{2} - 1}{2}

Simplify

\frac{\frac{3}{2} - 1}{2} : \frac{3 - 2}{4}

\frac{\frac{3}{2} - 1}{2}

Join

\frac{3}{2} - 1 : \frac{3 - 2}{2}

\frac{3}{2} - 1

Convert element to fraction:

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

= \frac{3}{2} - \frac{1 \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{3 - 1 \cdot 2}{2}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{3 - 2}{2}

= \frac{\frac{3 - 2}{2}}{2}

Apply the fraction rule:

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

= \frac{3 - 2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 - 2}{4}

= \frac{3 - 2}{4}

arccos (\frac{35}{77})

sin (- \frac{15 π}{4})

e^{0} cos (\frac{π}{3})

cos (π \frac{1}{2})

sin^{6} (\frac{π}{2})

What is the value of cos(105)cos(75) ?
The value of cos(105)cos(75) is (sqrt(3)-2)/4