What is the general solution for arcsin(x)=arcsin(2/5)+arcsin(3/5) ?

The general solution for arcsin(x)=arcsin(2/5)+arcsin(3/5) is x=(8+3sqrt(21))/(25)

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arcsin(x)=arcsin(2/5)+arcsin(3/5)

Solution

arcsin (x) = arcsin (\frac{2}{5}) + arcsin (\frac{3}{5})

Solution

x = \frac{8 + 3 21}{25}

Solution steps

arcsin (x) = arcsin (\frac{2}{5}) + arcsin (\frac{3}{5})

Apply trig inverse properties

arcsin (x) = arcsin (\frac{2}{5}) + arcsin (\frac{3}{5})

arcsin (x) = a \Rightarrow x = sin (a)

x = sin (arcsin (\frac{2}{5}) + arcsin (\frac{3}{5}))

sin (arcsin (\frac{2}{5}) + arcsin (\frac{3}{5})) = \frac{8 + 3 21}{25}

sin (arcsin (\frac{2}{5}) + arcsin (\frac{3}{5}))

Rewrite using trig identities:

sin (arcsin (\frac{2}{5})) cos (arcsin (\frac{3}{5})) + cos (arcsin (\frac{2}{5})) sin (arcsin (\frac{3}{5}))

sin (arcsin (\frac{2}{5}) + arcsin (\frac{3}{5}))

Use the Angle Sum identity:

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

= sin (arcsin (\frac{2}{5})) cos (arcsin (\frac{3}{5})) + cos (arcsin (\frac{2}{5})) sin (arcsin (\frac{3}{5}))

= sin (arcsin (\frac{2}{5})) cos (arcsin (\frac{3}{5})) + cos (arcsin (\frac{2}{5})) sin (arcsin (\frac{3}{5}))

Rewrite using trig identities:

sin (arcsin (\frac{2}{5})) = \frac{2}{5}

Use the following identity:

sin (arcsin (x)) = x

= \frac{2}{5}

Rewrite using trig identities:

cos (arcsin (\frac{3}{5})) = \frac{4}{5}

cos (arcsin (\frac{3}{5}))

Rewrite using trig identities:

cos (arcsin (\frac{3}{5})) = 1 - (\frac{3}{5})^{2}

Use the following identity:

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

= 1 - (\frac{3}{5})^{2}

= 1 - (\frac{3}{5})^{2}

Simplify

= \frac{4}{5}

Rewrite using trig identities:

cos (arcsin (\frac{2}{5})) = \frac{21}{5}

cos (arcsin (\frac{2}{5}))

Rewrite using trig identities:

cos (arcsin (\frac{2}{5})) = 1 - (\frac{2}{5})^{2}

Use the following identity:

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

= 1 - (\frac{2}{5})^{2}

= 1 - (\frac{2}{5})^{2}

Simplify

= \frac{21}{5}

Rewrite using trig identities:

sin (arcsin (\frac{3}{5})) = \frac{3}{5}

Use the following identity:

sin (arcsin (x)) = x

= \frac{3}{5}

= \frac{2}{5} \cdot \frac{4}{5} + \frac{21}{5} \cdot \frac{3}{5}

Simplify

\frac{2}{5} \cdot \frac{4}{5} + \frac{21}{5} \cdot \frac{3}{5} : \frac{8 + 3 21}{25}

\frac{2}{5} \cdot \frac{4}{5} + \frac{21}{5} \cdot \frac{3}{5}

\frac{2}{5} \cdot \frac{4}{5} = \frac{8}{25}

\frac{2}{5} \cdot \frac{4}{5}

Multiply fractions:

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

= \frac{2 \cdot 4}{5 \cdot 5}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{8}{5 \cdot 5}

Multiply the numbers:

5 \cdot 5 = 25

= \frac{8}{25}

\frac{21}{5} \cdot \frac{3}{5} = \frac{3 21}{25}

\frac{21}{5} \cdot \frac{3}{5}

Multiply fractions:

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

= \frac{21 \cdot 3}{5 \cdot 5}

Multiply the numbers:

5 \cdot 5 = 25

= \frac{3 21}{25}

= \frac{8}{25} + \frac{3 21}{25}

Apply rule

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

= \frac{8 + 3 21}{25}

= \frac{8 + 3 21}{25}

x = \frac{8 + 3 21}{25}

x = \frac{8 + 3 21}{25}

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Frequently Asked Questions (FAQ)

What is the general solution for arcsin(x)=arcsin(2/5)+arcsin(3/5) ?
The general solution for arcsin(x)=arcsin(2/5)+arcsin(3/5) is x=(8+3sqrt(21))/(25)