What is the value of sin(arcsin(1/2)+arctan(-4)) ?

The value of sin(arcsin(1/2)+arctan(-4)) is (sqrt(17))/(34)-(2sqrt(51))/(17)

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sin(arcsin(1/2)+arctan(-4))

Solution

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

Solution

\frac{17}{34} - \frac{2 51}{17}

Decimal Notation

- 0.71890 \dots

Solution steps

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

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 4) = - arctan (4)

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

Rewrite using trig identities:

sin (arcsin (\frac{1}{2})) cos (arctan (4)) - cos (arcsin (\frac{1}{2})) sin (arctan (4))

sin (arcsin (\frac{1}{2}) - arctan (4))

Use the Angle Difference identity:

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

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

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

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 (4)) = \frac{17}{17}

cos (arctan (4))

Rewrite using trig identities:

cos (arctan (4)) = \frac{1 + 4 ^{2}}{1 + 4 ^{2}}

Use the following identity:

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

= \frac{1 + 4 ^{2}}{1 + 4 ^{2}}

= \frac{1 + 4 ^{2}}{1 + 4 ^{2}}

Simplify

= \frac{17}{17}

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 (4)) = \frac{4 17}{17}

sin (arctan (4))

Rewrite using trig identities:

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

Use the following identity:

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

= \frac{4 1 + 4 ^{2}}{1 + 4 ^{2}}

= \frac{4 1 + 4 ^{2}}{1 + 4 ^{2}}

Simplify

= \frac{4 17}{17}

= \frac{1}{2} \cdot \frac{17}{17} - \frac{3}{2} \cdot \frac{4 17}{17}

Simplify

\frac{1}{2} \cdot \frac{17}{17} - \frac{3}{2} \cdot \frac{4 17}{17} : \frac{17}{34} - \frac{2 51}{17}

\frac{1}{2} \cdot \frac{17}{17} - \frac{3}{2} \cdot \frac{4 17}{17}

\frac{1}{2} \cdot \frac{17}{17} = \frac{17}{34}

\frac{1}{2} \cdot \frac{17}{17}

Multiply fractions:

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

= \frac{1 \cdot 17}{2 \cdot 17}

Multiply:

1 \cdot 17 = 17

= \frac{17}{2 \cdot 17}

Multiply the numbers:

2 \cdot 17 = 34

= \frac{17}{34}

\frac{3}{2} \cdot \frac{4 17}{17} = \frac{2 51}{17}

\frac{3}{2} \cdot \frac{4 17}{17}

Multiply fractions:

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

= \frac{3 \cdot 4 17}{2 \cdot 17}

Multiply the numbers:

2 \cdot 17 = 34

= \frac{4 3 17}{34}

Cancel the common factor:

2

= \frac{2 3 17}{17}

Simplify

2317 : 251

2317

Apply radical rule:

a b = a \cdot b

317 = 3 \cdot 17

= 2 3 \cdot 17

Multiply the numbers:

3 \cdot 17 = 51

= 251

= \frac{2 51}{17}

= \frac{17}{34} - \frac{2 51}{17}

= \frac{17}{34} - \frac{2 51}{17}

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Frequently Asked Questions (FAQ)

What is the value of sin(arcsin(1/2)+arctan(-4)) ?
The value of sin(arcsin(1/2)+arctan(-4)) is (sqrt(17))/(34)-(2sqrt(51))/(17)