What is the value of sin(arcsin(1/2)+arctan(-5)) ?

The value of sin(arcsin(1/2)+arctan(-5)) is (sqrt(26)-5sqrt(78))/(52)

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sin(arcsin(1/2)+arctan(-5))

Solution

sin (arcsin (\frac{1}{2}) + arctan (- 5))

Solution

\frac{26 - 5 78}{52}

Decimal Notation

- 0.75114 \dots

Solution steps

sin (arcsin (\frac{1}{2}) + arctan (- 5))

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 5) = - arctan (5)

= sin (arcsin (\frac{1}{2}) - arctan (5))

Rewrite using trig identities:

sin (arcsin (\frac{1}{2})) cos (arctan (5)) - cos (arcsin (\frac{1}{2})) sin (arctan (5))

sin (arcsin (\frac{1}{2}) - arctan (5))

Use the Angle Difference identity:

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

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

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

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 (5)) = \frac{26}{26}

cos (arctan (5))

Rewrite using trig identities:

cos (arctan (5)) = \frac{1 + 5 ^{2}}{1 + 5 ^{2}}

Use the following identity:

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

= \frac{1 + 5 ^{2}}{1 + 5 ^{2}}

= \frac{1 + 5 ^{2}}{1 + 5 ^{2}}

Simplify

= \frac{26}{26}

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 (5)) = \frac{5 26}{26}

sin (arctan (5))

Rewrite using trig identities:

sin (arctan (5)) = \frac{5 1 + 5 ^{2}}{1 + 5 ^{2}}

Use the following identity:

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

= \frac{5 1 + 5 ^{2}}{1 + 5 ^{2}}

= \frac{5 1 + 5 ^{2}}{1 + 5 ^{2}}

Simplify

= \frac{5 26}{26}

= \frac{1}{2} \cdot \frac{26}{26} - \frac{3}{2} \cdot \frac{5 26}{26}

Simplify

\frac{1}{2} \cdot \frac{26}{26} - \frac{3}{2} \cdot \frac{5 26}{26} : \frac{26 - 5 78}{52}

\frac{1}{2} \cdot \frac{26}{26} - \frac{3}{2} \cdot \frac{5 26}{26}

\frac{1}{2} \cdot \frac{26}{26} = \frac{26}{52}

\frac{1}{2} \cdot \frac{26}{26}

Multiply fractions:

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

= \frac{1 \cdot 26}{2 \cdot 26}

Multiply:

1 \cdot 26 = 26

= \frac{26}{2 \cdot 26}

Multiply the numbers:

2 \cdot 26 = 52

= \frac{26}{52}

\frac{3}{2} \cdot \frac{5 26}{26} = \frac{5 78}{52}

\frac{3}{2} \cdot \frac{5 26}{26}

Multiply fractions:

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

= \frac{3 \cdot 5 26}{2 \cdot 26}

Multiply the numbers:

2 \cdot 26 = 52

= \frac{5 3 26}{52}

Simplify

3 \cdot 526 : 578

3 \cdot 526

Apply radical rule:

a b = a \cdot b

326 = 3 \cdot 26

= 5 3 \cdot 26

Multiply the numbers:

3 \cdot 26 = 78

= 578

= \frac{5 78}{52}

= \frac{26}{52} - \frac{5 78}{52}

Apply rule

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

= \frac{26 - 5 78}{52}

= \frac{26 - 5 78}{52}

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Frequently Asked Questions (FAQ)

What is the value of sin(arcsin(1/2)+arctan(-5)) ?
The value of sin(arcsin(1/2)+arctan(-5)) is (sqrt(26)-5sqrt(78))/(52)