What is the value of (10^4*1.6*0.6)/(41)*sin(90+arctan(4/5)) ?

The value of (10^4*1.6*0.6)/(41)*sin(90+arctan(4/5)) is (48000sqrt(41))/(1681)

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Popular >

(10^41.60.6)/(41)*sin(90+arctan(4/5))

Solution

\frac{1 0 ^{4} \cdot 1.6 \cdot 0.6}{41} \cdot sin (9 0^{\circ} + arctan (\frac{4}{5}))

Solution

\frac{48000 41}{1681}

Decimal Notation

182.83757 \dots

Solution steps

\frac{1 0 ^{4} \cdot 1.6 \cdot 0.6}{41} sin (9 0^{\circ} + arctan (\frac{4}{5}))

= \frac{1 0 ^{4} \cdot \frac{8}{5} \cdot \frac{3}{5}}{41} sin (9 0^{\circ} + arctan (\frac{4}{5}))

Rewrite using trig identities:

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

sin (9 0^{\circ} + arctan (\frac{4}{5}))

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

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

= \frac{1 0 ^{4} \cdot \frac{8}{5} \cdot \frac{3}{5}}{41} (sin (9 0^{\circ}) cos (arctan (\frac{4}{5})) + cos (9 0^{\circ}) sin (arctan (\frac{4}{5})))

Simplify

= \frac{9600 ( sin ( 9 0 ^{\circ} ) cos ( arctan ( \frac{4}{5} ) ) + cos ( 9 0 ^{\circ} ) sin ( arctan ( \frac{4}{5} ) ) )}{41}

Use the following trivial identity:

sin (9 0^{\circ}) = 1

sin (9 0^{\circ})

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 1

Rewrite using trig identities:

cos (arctan (\frac{4}{5})) = \frac{5 41}{41}

cos (arctan (\frac{4}{5}))

Rewrite using trig identities:

cos (arctan (\frac{4}{5})) = \frac{1 + ( \frac{4}{5} ) ^{2}}{1 + ( \frac{4}{5} ) ^{2}}

Use the following identity:

cos (arctan (x)) = \frac{1 + x ^{2}}{1 + x ^{2}}

= \frac{1 + ( \frac{4}{5} ) ^{2}}{1 + ( \frac{4}{5} ) ^{2}}

= \frac{1 + ( \frac{4}{5} ) ^{2}}{1 + ( \frac{4}{5} ) ^{2}}

Simplify

= \frac{5 41}{41}

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

Rewrite using trig identities:

sin (arctan (\frac{4}{5})) = \frac{4 41}{41}

sin (arctan (\frac{4}{5}))

Rewrite using trig identities:

sin (arctan (\frac{4}{5})) = \frac{( \frac{4}{5} ) 1 + ( \frac{4}{5} ) ^{2}}{1 + ( \frac{4}{5} ) ^{2}}

Use the following identity:

sin (arctan (x)) = \frac{x 1 + x ^{2}}{1 + x ^{2}}

= \frac{( \frac{4}{5} ) 1 + ( \frac{4}{5} ) ^{2}}{1 + ( \frac{4}{5} ) ^{2}}

= \frac{\frac{4}{5} 1 + ( \frac{4}{5} ) ^{2}}{1 + ( \frac{4}{5} ) ^{2}}

Simplify

= \frac{4 41}{41}

= \frac{9600 ( 1 \cdot \frac{5 41}{41} + 0 \cdot \frac{4 41}{41} )}{41}

Simplify

\frac{9600 ( 1 \cdot \frac{5 41}{41} + 0 \cdot \frac{4 41}{41} )}{41} : \frac{48000 41}{1681}

\frac{9600 ( 1 \cdot \frac{5 41}{41} + 0 \cdot \frac{4 41}{41} )}{41}

9600 (1 \cdot \frac{5 41}{41} + 0 \cdot \frac{4 41}{41}) = 9600 \cdot \frac{5 41}{41}

9600 (1 \cdot \frac{5 41}{41} + 0 \cdot \frac{4 41}{41})

Multiply:

1 \cdot \frac{5 41}{41} = \frac{5 41}{41}

= 9600 (0 \cdot \frac{4 41}{41} + \frac{5 41}{41})

Apply rule

0 \cdot a = 0

= 9600 (\frac{5 41}{41} + 0)

\frac{5 41}{41} + 0 = \frac{5 41}{41}

= 9600 \cdot \frac{5 41}{41}

= \frac{9600 \cdot \frac{5 41}{41}}{41}

Multiply

9600 \cdot \frac{5 41}{41} : \frac{48000}{41}

9600 \cdot \frac{5 41}{41}

Multiply fractions:

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

= \frac{5 41 \cdot 9600}{41}

Multiply the numbers:

5 \cdot 9600 = 48000

= \frac{48000 41}{41}

Apply radical rule:

41 = 4 1^{\frac{1}{2}}

= \frac{48000 \cdot 4 1 ^{\frac{1}{2}}}{41}

Apply exponent rule:

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

\frac{4 1 ^{\frac{1}{2}}}{4 1 ^{1}} = \frac{1}{4 1 ^{1 - \frac{1}{2}}}

= \frac{48000}{4 1 ^{1 - \frac{1}{2}}}

Subtract the numbers:

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

= \frac{48000}{4 1 ^{\frac{1}{2}}}

Apply radical rule:

4 1^{\frac{1}{2}} = 41

= \frac{48000}{41}

= \frac{\frac{48000}{41}}{41}

Apply the fraction rule:

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

= \frac{48000}{41 \cdot 41}

Rationalize

\frac{48000}{41 41} : \frac{48000 41}{1681}

\frac{48000}{41 41}

Multiply by the conjugate

\frac{41}{41}

= \frac{48000 41}{41 \cdot 41 41}

41 \cdot 4141 = 1681

41 \cdot 4141

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

414141 = 41 \cdot 4 1^{\frac{1}{2}} \cdot 4 1^{\frac{1}{2}} = 4 1^{1 + \frac{1}{2} + \frac{1}{2}}

= 4 1^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

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

= 4 1^{1 + 2 \cdot \frac{1}{2}}

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

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 4 1^{1 + 1}

Add the numbers:

1 + 1 = 2

= 4 1^{2}

4 1^{2} = 1681

= 1681

= \frac{48000 41}{1681}

= \frac{48000 41}{1681}

= \frac{48000 41}{1681}

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Frequently Asked Questions (FAQ)

What is the value of (10^4*1.6*0.6)/(41)*sin(90+arctan(4/5)) ?
The value of (10^4*1.6*0.6)/(41)*sin(90+arctan(4/5)) is (48000sqrt(41))/(1681)