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

The value of sin(arcsin(1/2)+arctan(-3)) is (sqrt(10)-3sqrt(30))/(20)

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Popular >

sin(arcsin(1/2)+arctan(-3))

Solution

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

Solution

\frac{10 - 3 30}{20}

Decimal Notation

- 0.66346 \dots

Solution steps

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

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 3) = - arctan (3)

= sin (arcsin (\frac{1}{2}) - arctan (3))

Rewrite using trig identities:

sin (arcsin (\frac{1}{2})) cos (arctan (3)) - cos (arcsin (\frac{1}{2})) sin (arctan (3))

sin (arcsin (\frac{1}{2}) - arctan (3))

Use the Angle Difference identity:

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

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

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

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 (3)) = \frac{10}{10}

cos (arctan (3))

Rewrite using trig identities:

cos (arctan (3)) = \frac{1 + 3 ^{2}}{1 + 3 ^{2}}

Use the following identity:

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

= \frac{1 + 3 ^{2}}{1 + 3 ^{2}}

= \frac{1 + 3 ^{2}}{1 + 3 ^{2}}

Simplify

= \frac{10}{10}

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 (3)) = \frac{3 10}{10}

sin (arctan (3))

Rewrite using trig identities:

sin (arctan (3)) = \frac{3 1 + 3 ^{2}}{1 + 3 ^{2}}

Use the following identity:

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

= \frac{3 1 + 3 ^{2}}{1 + 3 ^{2}}

= \frac{3 1 + 3 ^{2}}{1 + 3 ^{2}}

Simplify

= \frac{3 10}{10}

= \frac{1}{2} \cdot \frac{10}{10} - \frac{3}{2} \cdot \frac{3 10}{10}

Simplify

\frac{1}{2} \cdot \frac{10}{10} - \frac{3}{2} \cdot \frac{3 10}{10} : \frac{10 - 3 30}{20}

\frac{1}{2} \cdot \frac{10}{10} - \frac{3}{2} \cdot \frac{3 10}{10}

\frac{1}{2} \cdot \frac{10}{10} = \frac{10}{20}

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

Multiply fractions:

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

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

Multiply:

1 \cdot 10 = 10

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

Multiply the numbers:

2 \cdot 10 = 20

= \frac{10}{20}

\frac{3}{2} \cdot \frac{3 10}{10} = \frac{3 30}{20}

\frac{3}{2} \cdot \frac{3 10}{10}

Multiply fractions:

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

= \frac{3 \cdot 3 10}{2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{3 3 10}{20}

Simplify

3 \cdot 310 : 330

3 \cdot 310

Apply radical rule:

a b = a \cdot b

310 = 3 \cdot 10

= 3 3 \cdot 10

Multiply the numbers:

3 \cdot 10 = 30

= 330

= \frac{3 30}{20}

= \frac{10}{20} - \frac{3 30}{20}

Apply rule

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

= \frac{10 - 3 30}{20}

= \frac{10 - 3 30}{20}

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Frequently Asked Questions (FAQ)

What is the value of sin(arcsin(1/2)+arctan(-3)) ?
The value of sin(arcsin(1/2)+arctan(-3)) is (sqrt(10)-3sqrt(30))/(20)