What is the value of 2arcsec(2)+arcsin(1/2) ?

The value of 2arcsec(2)+arcsin(1/2) is (5pi)/6

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2arcsec(2)+arcsin(1/2)

Solution

2 arcsec (2) + arcsin (\frac{1}{2})

Solution

\frac{5 π}{6}

Decimal Notation

150

Solution steps

2 arcsec (2) + arcsin (\frac{1}{2})

Use the following trivial identity:

arcsec (2) = \frac{π}{3}

arcsec (2)

x 1 \frac{2 3}{3} 22 - 2 - 2 - \frac{2 3}{3} - 1 arcsec (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} π arcsec (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} 18 0^{\circ}

= \frac{π}{3}

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{6}

arcsin (\frac{1}{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}{6}

= 2 \cdot \frac{π}{3} + \frac{π}{6}

Simplify

2 \cdot \frac{π}{3} + \frac{π}{6} : \frac{5 π}{6}

2 \cdot \frac{π}{3} + \frac{π}{6}

Multiply

2 \cdot \frac{π}{3} : \frac{2 π}{3}

2 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 2}{3}

= \frac{2 π}{3} + \frac{π}{6}

Least Common Multiplier of

3, 6 : 6

3, 6

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

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

3

6

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 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}{3} :

multiply the denominator and numerator by

2

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

= \frac{4 π}{6} + \frac{π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{4 π + π}{6}

Add similar elements:

4 π + π = 5 π

= \frac{5 π}{6}

= \frac{5 π}{6}

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Frequently Asked Questions (FAQ)

What is the value of 2arcsec(2)+arcsin(1/2) ?
The value of 2arcsec(2)+arcsin(1/2) is (5pi)/6