What is the value of cos(arcsin(4/5)+arccos(3/5)) ?

The value of cos(arcsin(4/5)+arccos(3/5)) is -7/25

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cos(arcsin(4/5)+arccos(3/5))

Solution

cos (arcsin (\frac{4}{5}) + arccos (\frac{3}{5}))

Solution

- \frac{7}{25}

Decimal Notation

- 0.28

Solution steps

cos (arcsin (\frac{4}{5}) + arccos (\frac{3}{5}))

Rewrite using trig identities:

cos (arcsin (\frac{4}{5})) cos (arccos (\frac{3}{5})) - sin (arcsin (\frac{4}{5})) sin (arccos (\frac{3}{5}))

cos (arcsin (\frac{4}{5}) + arccos (\frac{3}{5}))

Use the Angle Sum identity:

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

= cos (arcsin (\frac{4}{5})) cos (arccos (\frac{3}{5})) - sin (arcsin (\frac{4}{5})) sin (arccos (\frac{3}{5}))

= cos (arcsin (\frac{4}{5})) cos (arccos (\frac{3}{5})) - sin (arcsin (\frac{4}{5})) sin (arccos (\frac{3}{5}))

Rewrite using trig identities:

cos (arcsin (\frac{4}{5})) = \frac{3}{5}

cos (arcsin (\frac{4}{5}))

Rewrite using trig identities:

cos (arcsin (\frac{4}{5})) = 1 - (\frac{4}{5})^{2}

Use the following identity:

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

= 1 - (\frac{4}{5})^{2}

= 1 - (\frac{4}{5})^{2}

Simplify

= \frac{3}{5}

Rewrite using trig identities:

cos (arccos (\frac{3}{5})) = \frac{3}{5}

Use the following identity:

cos (arccos (x)) = x

= \frac{3}{5}

Rewrite using trig identities:

sin (arcsin (\frac{4}{5})) = \frac{4}{5}

Use the following identity:

sin (arcsin (x)) = x

= \frac{4}{5}

Rewrite using trig identities:

sin (arccos (\frac{3}{5})) = \frac{4}{5}

sin (arccos (\frac{3}{5}))

Rewrite using trig identities:

sin (arccos (\frac{3}{5})) = 1 - (\frac{3}{5})^{2}

Use the following identity:

sin (arccos (x)) = 1 - x^{2}

= 1 - (\frac{3}{5})^{2}

= 1 - (\frac{3}{5})^{2}

Simplify

= \frac{4}{5}

= \frac{3}{5} \cdot \frac{3}{5} - \frac{4}{5} \cdot \frac{4}{5}

Simplify

\frac{3}{5} \cdot \frac{3}{5} - \frac{4}{5} \cdot \frac{4}{5} : - \frac{7}{25}

\frac{3}{5} \cdot \frac{3}{5} - \frac{4}{5} \cdot \frac{4}{5}

\frac{3}{5} \cdot \frac{3}{5} = \frac{9}{25}

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

Multiply fractions:

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

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

Multiply the numbers:

3 \cdot 3 = 9

= \frac{9}{5 \cdot 5}

Multiply the numbers:

5 \cdot 5 = 25

= \frac{9}{25}

\frac{4}{5} \cdot \frac{4}{5} = \frac{16}{25}

\frac{4}{5} \cdot \frac{4}{5}

Multiply fractions:

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

= \frac{4 \cdot 4}{5 \cdot 5}

Multiply the numbers:

4 \cdot 4 = 16

= \frac{16}{5 \cdot 5}

Multiply the numbers:

5 \cdot 5 = 25

= \frac{16}{25}

= \frac{9}{25} - \frac{16}{25}

Apply rule

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

= \frac{9 - 16}{25}

Subtract the numbers:

9 - 16 = - 7

= \frac{- 7}{25}

Apply the fraction rule:

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

= - \frac{7}{25}

= - \frac{7}{25}

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Frequently Asked Questions (FAQ)

What is the value of cos(arcsin(4/5)+arccos(3/5)) ?
The value of cos(arcsin(4/5)+arccos(3/5)) is -7/25