What is the value of cos(arcsin(1/2)+arccos(4/5)) ?

The value of cos(arcsin(1/2)+arccos(4/5)) is (2sqrt(3))/5-3/10

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cos(arcsin(1/2)+arccos(4/5))

Solution

cos (arcsin (\frac{1}{2}) + arccos (\frac{4}{5}))

Solution

\frac{2 3}{5} - \frac{3}{10}

Decimal Notation

0.39282 \dots

Solution steps

cos (arcsin (\frac{1}{2}) + arccos (\frac{4}{5}))

Rewrite using trig identities:

cos (arcsin (\frac{1}{2})) cos (arccos (\frac{4}{5})) - sin (arcsin (\frac{1}{2})) sin (arccos (\frac{4}{5}))

cos (arcsin (\frac{1}{2}) + arccos (\frac{4}{5}))

Use the Angle Sum identity:

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

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

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

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:

cos (arccos (\frac{4}{5})) = \frac{4}{5}

Use the following identity:

cos (arccos (x)) = x

= \frac{4}{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:

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

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{3}{5}

= \frac{3}{2} \cdot \frac{4}{5} - \frac{1}{2} \cdot \frac{3}{5}

Simplify

\frac{3}{2} \cdot \frac{4}{5} - \frac{1}{2} \cdot \frac{3}{5} : \frac{2 3}{5} - \frac{3}{10}

\frac{3}{2} \cdot \frac{4}{5} - \frac{1}{2} \cdot \frac{3}{5}

\frac{3}{2} \cdot \frac{4}{5} = \frac{2 3}{5}

\frac{3}{2} \cdot \frac{4}{5}

Cross-cancel common factor:

2

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 5 = 5

= \frac{2 3}{5}

\frac{1}{2} \cdot \frac{3}{5} = \frac{3}{10}

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 3 = 3

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

Multiply the numbers:

2 \cdot 5 = 10

= \frac{3}{10}

= \frac{2 3}{5} - \frac{3}{10}

= \frac{2 3}{5} - \frac{3}{10}

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Frequently Asked Questions (FAQ)

What is the value of cos(arcsin(1/2)+arccos(4/5)) ?
The value of cos(arcsin(1/2)+arccos(4/5)) is (2sqrt(3))/5-3/10