What is the value of arccos(0)+arcsin((sqrt(2))/2)+arctan(-1)+arccot(-(sqrt(3))/3) ?

The value of arccos(0)+arcsin((sqrt(2))/2)+arctan(-1)+arccot(-(sqrt(3))/3) is (7pi)/6

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Popular >

arccos(0)+arcsin((sqrt(2))/2)+arctan(-1)+arccot(-(sqrt(3))/3)

Solution

arccos (0) + arcsin (\frac{2}{2}) + arctan (- 1) + arccot (- \frac{3}{3})

Solution

\frac{7 π}{6}

Decimal Notation

210

Solution steps

arccos (0) + arcsin (\frac{2}{2}) + arctan (- 1) + arccot (- \frac{3}{3})

Use the following trivial identity:

arccos (0) = \frac{π}{2}

arccos (0)

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= \frac{π}{2}

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

arcsin (\frac{2}{2})

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{4}

Rewrite using trig identities:

arctan (- 1) = - \frac{π}{4}

arctan (- 1)

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 1) = - arctan (1)

= - arctan (1)

Use the following trivial identity:

arctan (1) = \frac{π}{4}

arctan (1)

x 0 \frac{3}{3} 13 arctan (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} arctan (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ}

= \frac{π}{4}

= - \frac{π}{4}

Use the following trivial identity:

arccot (- \frac{3}{3}) = \frac{2 π}{3}

arccot (- \frac{3}{3})

x - 3 - 1 - \frac{3}{3} 0 \frac{3}{3} 13 arccot (x) \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} arccot (x) 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ}

= \frac{2 π}{3}

= \frac{π}{2} + \frac{π}{4} - \frac{π}{4} + \frac{2 π}{3}

Simplify

\frac{π}{2} + \frac{π}{4} - \frac{π}{4} + \frac{2 π}{3} : \frac{7 π}{6}

\frac{π}{2} + \frac{π}{4} - \frac{π}{4} + \frac{2 π}{3}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= \frac{π}{2} + \frac{2 π}{3}

Least Common Multiplier of

2, 3 : 6

2, 3

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Multiply each factor the greatest number of times it occurs in either

2

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{π}{2} :

multiply the denominator and numerator by

3

\frac{π}{2} = \frac{π 3}{2 \cdot 3} = \frac{π 3}{6}

For

\frac{2 π}{3} :

multiply the denominator and numerator by

2

\frac{2 π}{3} = \frac{2 π 2}{3 \cdot 2} = \frac{4 π}{6}

= \frac{π 3}{6} + \frac{4 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 + 4 π}{6}

Add similar elements:

3 π + 4 π = 7 π

= \frac{7 π}{6}

= \frac{7 π}{6}

Popular Examples

6sin((3pi)/4)

6 sin (\frac{3 π}{4})

3cos(1/2)

3 cos (\frac{1}{2})

cos(pi/5)cos(pi/3)-sin(pi/5)sin(pi/3)

cos (\frac{π}{5}) cos (\frac{π}{3}) - sin (\frac{π}{5}) sin (\frac{π}{3})

cos(pi-1)

cos (π - 1)

arctan(90/60)

arctan (\frac{90}{60})

Frequently Asked Questions (FAQ)

What is the value of arccos(0)+arcsin((sqrt(2))/2)+arctan(-1)+arccot(-(sqrt(3))/3) ?
The value of arccos(0)+arcsin((sqrt(2))/2)+arctan(-1)+arccot(-(sqrt(3))/3) is (7pi)/6