What is the value of 10cos(75)cos(15) ?

The value of 10cos(75)cos(15) is 5/2

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10cos(75)cos(15)

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

\frac{5}{2}

Decimal Notation

2.5

Solution steps

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

Rewrite using trig identities:

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

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

Use the Product to Sum identity:

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

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

Simplify

= 5 (cos (6 0^{\circ}) + cos (9 0^{\circ}))

= 5 (cos (6 0^{\circ}) + cos (9 0^{\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 (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

= 5 (\frac{1}{2} + 0)

Simplify

5 (\frac{1}{2} + 0) : \frac{5}{2}

5 (\frac{1}{2} + 0)

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

= 5 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 5}{2}

Multiply the numbers:

1 \cdot 5 = 5

= \frac{5}{2}

= \frac{5}{2}

cos^{2} (\frac{5 π}{6}) - 2 sin (\frac{5 π}{6})

- sin (27 0^{\circ})

arcsin (sin (- \frac{7 π}{4}))

\frac{25}{sin ( 2 7 ^{\circ} )}

5 \cdot sin (3 0^{\circ})