What is the value of 31cos(45) ?

The value of 31cos(45) is (31sqrt(2))/2

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31cos(45)

31 cos (4 5^{\circ})

\frac{31 2}{2}

Decimal Notation

21.92031 \dots

Solution steps

31 cos (4 5^{\circ})

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}

= 31 \cdot \frac{2}{2}

Simplify

31 \cdot \frac{2}{2} : \frac{31 2}{2}

31 \cdot \frac{2}{2}

Multiply fractions:

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

= \frac{2 \cdot 31}{2}

Apply radical rule:

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

= \frac{31 \cdot 2 ^{\frac{1}{2}}}{2}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{31}{2 ^{1 - \frac{1}{2}}}

Subtract the numbers:

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

= \frac{31}{2 ^{\frac{1}{2}}}

Apply radical rule:

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

= \frac{31}{2}

Rationalize

\frac{31}{2} : \frac{31 2}{2}

\frac{31}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{31 2}{2 2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{31 2}{2}

= \frac{31 2}{2}

= \frac{31 2}{2}

sec (49.9 6^{\circ})

tan (arccos (\frac{7}{11}))

sec (arctan (\frac{3 2}{2}))

\frac{10 0 ^{2} ( 2 sin ( 3 0 ^{\circ} ) )}{9.8} - 999.984741

\frac{sin ( 8 0 ^{\circ} ) + sin ( 2 0 ^{\circ} )}{cos ( 8 0 ^{\circ} ) + cos ( 2 0 ^{\circ} )}