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

The value of sin(arctan(-1)+arcsin(-(sqrt(2))/2)) is -1

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

Solution

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

Solution

- 1

Solution steps

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

Rewrite using trig identities:

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

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

Use the Angle Sum identity:

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

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

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2}{2}) = - arcsin (\frac{2}{2})

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

Use the following property:

sin (- x) = - sin (x)

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

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

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 1) = - arctan (1)

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

Use the following property:

cos (- x) = cos (x)

cos (- arctan (1)) = cos (arctan (1))

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

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2}{2}) = - arcsin (\frac{2}{2})

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

Use the following property:

cos (- x) = cos (x)

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

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

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 1) = - arctan (1)

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

Use the following property:

sin (- x) = - sin (x)

sin (- arctan (1)) = - sin (arctan (1))

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

Simplify

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

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

Rewrite using trig identities:

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

sin (arctan (1))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{2}{2}

Rewrite using trig identities:

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

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{1}{2}

Rewrite using trig identities:

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

cos (arctan (1))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{2}{2}

Rewrite using trig identities:

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

Use the following identity:

sin (arcsin (x)) = x

= \frac{2}{2}

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

Simplify

- \frac{2}{2} \frac{1}{2} - \frac{2}{2} \cdot \frac{2}{2} : - 1

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

\frac{2}{2} \frac{1}{2} = \frac{1}{2}

\frac{2}{2} \frac{1}{2}

Multiply fractions:

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

= \frac{2 \frac{1}{2}}{2}

\frac{1}{2} = \frac{1}{2}

\frac{1}{2}

Apply radical rule:

n \frac{a}{b} = \frac{n a}{n b},

assuming

a \geq 0, b \geq 0

= \frac{1}{2}

Apply rule

1 = 1

= \frac{1}{2}

= \frac{2 \frac{1}{2}}{2}

Multiply

2 \frac{1}{2} : 1

2 \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= \frac{1}{2}

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

\frac{2}{2} \cdot \frac{2}{2}

Multiply fractions:

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

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

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

= - \frac{1}{2} - \frac{1}{2}

Apply rule

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

= \frac{- 1 - 1}{2}

Subtract the numbers:

- 1 - 1 = - 2

= \frac{- 2}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= - 1

= - 1

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Frequently Asked Questions (FAQ)

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