What is the value of sin(arcsin(2/3)+arccos(1/2)) ?

The value of sin(arcsin(2/3)+arccos(1/2)) is 1/3+(sqrt(15))/6

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sin(arcsin(2/3)+arccos(1/2))

Solution

sin (arcsin (\frac{2}{3}) + arccos (\frac{1}{2}))

Solution

\frac{1}{3} + \frac{15}{6}

Decimal Notation

0.97883 \dots

Solution steps

sin (arcsin (\frac{2}{3}) + arccos (\frac{1}{2}))

Rewrite using trig identities:

sin (arcsin (\frac{2}{3})) cos (arccos (\frac{1}{2})) + cos (arcsin (\frac{2}{3})) sin (arccos (\frac{1}{2}))

sin (arcsin (\frac{2}{3}) + arccos (\frac{1}{2}))

Use the Angle Sum identity:

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

= sin (arcsin (\frac{2}{3})) cos (arccos (\frac{1}{2})) + cos (arcsin (\frac{2}{3})) sin (arccos (\frac{1}{2}))

= sin (arcsin (\frac{2}{3})) cos (arccos (\frac{1}{2})) + cos (arcsin (\frac{2}{3})) sin (arccos (\frac{1}{2}))

Rewrite using trig identities:

sin (arcsin (\frac{2}{3})) = \frac{2}{3}

Use the following identity:

sin (arcsin (x)) = x

= \frac{2}{3}

Rewrite using trig identities:

cos (arccos (\frac{1}{2})) = \frac{1}{2}

Use the following identity:

cos (arccos (x)) = x

= \frac{1}{2}

Rewrite using trig identities:

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

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{5}{3}

Rewrite using trig identities:

sin (arccos (\frac{1}{2})) = \frac{3}{2}

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

Rewrite using trig identities:

sin (arccos (\frac{1}{2})) = 1 - (\frac{1}{2})^{2}

Use the following identity:

sin (arccos (x)) = 1 - x^{2}

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

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

Simplify

= \frac{3}{2}

= \frac{2}{3} \cdot \frac{1}{2} + \frac{5}{3} \cdot \frac{3}{2}

Simplify

\frac{2}{3} \cdot \frac{1}{2} + \frac{5}{3} \cdot \frac{3}{2} : \frac{1}{3} + \frac{15}{6}

\frac{2}{3} \cdot \frac{1}{2} + \frac{5}{3} \cdot \frac{3}{2}

\frac{2}{3} \cdot \frac{1}{2} = \frac{1}{3}

\frac{2}{3} \cdot \frac{1}{2}

Cross-cancel:

2

= \frac{1}{3}

\frac{5}{3} \cdot \frac{3}{2} = \frac{15}{6}

\frac{5}{3} \cdot \frac{3}{2}

Multiply fractions:

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

= \frac{5 3}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{5 3}{6}

Simplify

53 : 15

53

Apply radical rule:

a b = a \cdot b

53 = 5 \cdot 3

= 5 \cdot 3

Multiply the numbers:

5 \cdot 3 = 15

= 15

= \frac{15}{6}

= \frac{1}{3} + \frac{15}{6}

= \frac{1}{3} + \frac{15}{6}

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Frequently Asked Questions (FAQ)

What is the value of sin(arcsin(2/3)+arccos(1/2)) ?
The value of sin(arcsin(2/3)+arccos(1/2)) is 1/3+(sqrt(15))/6