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

The value of 1/2 (arcsin(1)+1/2 sin(2arcsin(1))) is (arcsin(1))/2

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1/2 (arcsin(1)+1/2 sin(2arcsin(1)))

Solution

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

Solution

\frac{arcsin ( 1 )}{2}

Decimal Notation

0.78539 \dots

Solution steps

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

\frac{1}{2} (arcsin (1) + \frac{1}{2} sin (2 arcsin (1))) = \frac{2 arcsin ( 1 ) + sin ( 2 arcsin ( 1 ) )}{4}

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

Multiply fractions:

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

= \frac{1 \cdot ( arcsin ( 1 ) + \frac{1}{2} sin ( 2 arcsin ( 1 ) ) )}{2}

1 \cdot (arcsin (1) + \frac{1}{2} sin (2 arcsin (1))) = arcsin (1) + \frac{1}{2} sin (2 arcsin (1))

1 \cdot (arcsin (1) + \frac{1}{2} sin (2 arcsin (1)))

Multiply:

1 \cdot (arcsin (1) + \frac{1}{2} sin (2 arcsin (1))) = (arcsin (1) + \frac{1}{2} sin (2 arcsin (1)))

= (arcsin (1) + \frac{1}{2} sin (2 arcsin (1)))

Remove parentheses:

(a) = a

= arcsin (1) + \frac{1}{2} sin (2 arcsin (1))

= \frac{arcsin ( 1 ) + \frac{1}{2} sin ( 2 arcsin ( 1 ) )}{2}

\frac{1}{2} sin (2 arcsin (1)) = \frac{sin ( 2 arcsin ( 1 ) )}{2}

\frac{1}{2} sin (2 arcsin (1))

Multiply fractions:

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

= \frac{1 \cdot sin ( 2 arcsin ( 1 ) )}{2}

Multiply:

1 \cdot sin (2 arcsin (1)) = sin (2 arcsin (1))

= \frac{sin ( 2 arcsin ( 1 ) )}{2}

= \frac{arcsin ( 1 ) + \frac{s i n ( 2 a r c s i n ( 1 ) )}{2}}{2}

Join

arcsin (1) + \frac{sin ( 2 arcsin ( 1 ) )}{2} : \frac{2 arcsin ( 1 ) + sin ( 2 arcsin ( 1 ) )}{2}

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

Convert element to fraction:

arcsin (1) = \frac{arcsin ( 1 ) 2}{2}

= \frac{arcsin ( 1 ) \cdot 2}{2} + \frac{sin ( 2 arcsin ( 1 ) )}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{arcsin ( 1 ) \cdot 2 + sin ( 2 arcsin ( 1 ) )}{2}

= \frac{\frac{2 a r c s i n ( 1 ) + s i n ( 2 a r c s i n ( 1 ) )}{2}}{2}

Apply the fraction rule:

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

= \frac{arcsin ( 1 ) \cdot 2 + sin ( 2 arcsin ( 1 ) )}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 arcsin ( 1 ) + sin ( 2 arcsin ( 1 ) )}{4}

= \frac{2 arcsin ( 1 ) + sin ( 2 arcsin ( 1 ) )}{4}

Rewrite using trig identities:

sin (2 arcsin (1)) = 2 sin (arcsin (1)) cos (arcsin (1))

sin (2 arcsin (1))

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= 2 sin (arcsin (1)) cos (arcsin (1))

= \frac{2 arcsin ( 1 ) + 2 sin ( arcsin ( 1 ) ) cos ( arcsin ( 1 ) )}{4}

Rewrite using trig identities:

sin (arcsin (1)) = 1

Use the following identity:

sin (arcsin (x)) = x

= 1

Rewrite using trig identities:

cos (arcsin (1)) = 0

cos (arcsin (1))

Rewrite using trig identities:

cos (arcsin (1)) = 1 - 1^{2}

Use the following identity:

cos (arcsin (x)) = 1 - x^{2}

= 1 - 1^{2}

= 1 - 1^{2}

Simplify

= 0

= \frac{2 arcsin ( 1 ) + 2 \cdot 1 \cdot 0}{4}

Simplify

\frac{2 arcsin ( 1 ) + 2 \cdot 1 \cdot 0}{4} : \frac{arcsin ( 1 )}{2}

\frac{2 arcsin ( 1 ) + 2 \cdot 1 \cdot 0}{4}

2 arcsin (1) + 2 \cdot 1 \cdot 0 = 2 arcsin (1)

2 arcsin (1) + 2 \cdot 1 \cdot 0

Apply rule

0 \cdot a = 0

= 2 arcsin (1) + 0

2 arcsin (1) + 0 = 2 arcsin (1)

= 2 arcsin (1)

= \frac{2 arcsin ( 1 )}{4}

Cancel the common factor:

2

= \frac{arcsin ( 1 )}{2}

= \frac{arcsin ( 1 )}{2}

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Frequently Asked Questions (FAQ)

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