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

The value of sin(arcsin(1/2)+arctan(-2)) is (sqrt(5)}{10}-\frac{sqrt(15))/5

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

Solution

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

Solution

\frac{5}{10} - \frac{15}{5}

Decimal Notation

- 0.55098 \dots

Solution steps

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

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 2) = - arctan (2)

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

Rewrite using trig identities:

sin (arcsin (\frac{1}{2})) cos (arctan (2)) - cos (arcsin (\frac{1}{2})) sin (arctan (2))

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

Use the Angle Difference identity:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= sin (arcsin (\frac{1}{2})) cos (arctan (2)) - cos (arcsin (\frac{1}{2})) sin (arctan (2))

= sin (arcsin (\frac{1}{2})) cos (arctan (2)) - cos (arcsin (\frac{1}{2})) sin (arctan (2))

Rewrite using trig identities:

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

Use the following identity:

sin (arcsin (x)) = x

= \frac{1}{2}

Rewrite using trig identities:

cos (arctan (2)) = \frac{5}{5}

cos (arctan (2))

Rewrite using trig identities:

cos (arctan (2)) = \frac{1 + 2 ^{2}}{1 + 2 ^{2}}

Use the following identity:

cos (arctan (x)) = \frac{1 + x ^{2}}{1 + x ^{2}}

= \frac{1 + 2 ^{2}}{1 + 2 ^{2}}

= \frac{1 + 2 ^{2}}{1 + 2 ^{2}}

Simplify

= \frac{5}{5}

Rewrite using trig identities:

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

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{3}{2}

Rewrite using trig identities:

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

sin (arctan (2))

Rewrite using trig identities:

sin (arctan (2)) = \frac{2 1 + 2 ^{2}}{1 + 2 ^{2}}

Use the following identity:

sin (arctan (x)) = \frac{x 1 + x ^{2}}{1 + x ^{2}}

= \frac{2 1 + 2 ^{2}}{1 + 2 ^{2}}

= \frac{2 1 + 2 ^{2}}{1 + 2 ^{2}}

Simplify

= \frac{2 5}{5}

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

Simplify

\frac{1}{2} \cdot \frac{5}{5} - \frac{3}{2} \cdot \frac{2 5}{5} : \frac{5}{10} - \frac{15}{5}

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

\frac{1}{2} \cdot \frac{5}{5} = \frac{5}{10}

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

Multiply fractions:

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

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

Multiply:

1 \cdot 5 = 5

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

Multiply the numbers:

2 \cdot 5 = 10

= \frac{5}{10}

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

= \frac{3 5}{5}

Simplify

35 : 15

35

Apply radical rule:

a b = a \cdot b

35 = 3 \cdot 5

= 3 \cdot 5

Multiply the numbers:

3 \cdot 5 = 15

= 15

= \frac{15}{5}

= \frac{5}{10} - \frac{15}{5}

= \frac{5}{10} - \frac{15}{5}

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Frequently Asked Questions (FAQ)

What is the value of sin(arcsin(1/2)+arctan(-2)) ?
The value of sin(arcsin(1/2)+arctan(-2)) is (sqrt(5)}{10}-\frac{sqrt(15))/5