What is the value of cos(75)cos(15) ?

The value of cos(75)cos(15) is 1/4

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cos(75)cos(15)

cos (7 5^{\circ}) cos (1 5^{\circ})

\frac{1}{4}

Decimal Notation

0.25

Solution steps

cos (7 5^{\circ}) cos (1 5^{\circ})

Rewrite using trig identities:

\frac{cos ( 6 0 ^{\circ} ) + cos ( 9 0 ^{\circ} )}{2}

cos (1 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 (1 5^{\circ} - 7 5^{\circ}) + cos (1 5^{\circ} + 7 5^{\circ}))

Simplify

= \frac{cos ( - 6 0 ^{\circ} ) + cos ( 9 0 ^{\circ} )}{2}

Use the following property:

cos (- x) = cos (x)

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

= \frac{cos ( 6 0 ^{\circ} ) + cos ( 9 0 ^{\circ} )}{2}

= \frac{cos ( 6 0 ^{\circ} ) + cos ( 9 0 ^{\circ} )}{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}

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

= \frac{\frac{1}{2} + 0}{2}

Simplify

\frac{\frac{1}{2} + 0}{2} : \frac{1}{4}

\frac{\frac{1}{2} + 0}{2}

\frac{1}{2} + 0 = \frac{1}{2}

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

= \frac{1}{4}

tan (4 5^{\circ} - 3 0^{\circ})

sec (1 1^{\circ})

csc (- 45 0^{\circ})

arccos (0.72)

arctan (\frac{10}{6})