What is the value of cos(arcsin(1/3)-arctan(1/2)) ?

The value of cos(arcsin(1/3)-arctan(1/2)) is (4sqrt(10)+sqrt(5))/(15)

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cos(arcsin(1/3)-arctan(1/2))

Solution

cos (arcsin (\frac{1}{3}) - arctan (\frac{1}{2}))

Solution

\frac{4 10 + 5}{15}

Decimal Notation

0.99234 \dots

Solution steps

cos (arcsin (\frac{1}{3}) - arctan (\frac{1}{2}))

Rewrite using trig identities:

cos (arcsin (\frac{1}{3})) cos (arctan (\frac{1}{2})) + sin (arcsin (\frac{1}{3})) sin (arctan (\frac{1}{2}))

cos (arcsin (\frac{1}{3}) - arctan (\frac{1}{2}))

Use the Angle Difference identity:

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

= cos (arcsin (\frac{1}{3})) cos (arctan (\frac{1}{2})) + sin (arcsin (\frac{1}{3})) sin (arctan (\frac{1}{2}))

= cos (arcsin (\frac{1}{3})) cos (arctan (\frac{1}{2})) + sin (arcsin (\frac{1}{3})) sin (arctan (\frac{1}{2}))

Rewrite using trig identities:

cos (arcsin (\frac{1}{3})) = \frac{2 2}{3}

cos (arcsin (\frac{1}{3}))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{2 2}{3}

Rewrite using trig identities:

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

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{2 5}{5}

Rewrite using trig identities:

sin (arcsin (\frac{1}{3})) = \frac{1}{3}

Use the following identity:

sin (arcsin (x)) = x

= \frac{1}{3}

Rewrite using trig identities:

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

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{5}{5}

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

Simplify

\frac{2 2}{3} \cdot \frac{2 5}{5} + \frac{1}{3} \cdot \frac{5}{5} : \frac{4 10 + 5}{15}

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

\frac{2 2}{3} \cdot \frac{2 5}{5} = \frac{4 10}{15}

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

Multiply fractions:

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

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

Multiply the numbers:

2 \cdot 2 = 4

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

Multiply the numbers:

3 \cdot 5 = 15

= \frac{4 2 5}{15}

Simplify

425 : 410

425

Apply radical rule:

a b = a \cdot b

25 = 2 \cdot 5

= 4 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 410

= \frac{4 10}{15}

\frac{1}{3} \cdot \frac{5}{5} = \frac{5}{15}

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

Multiply fractions:

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

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

Multiply:

1 \cdot 5 = 5

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

Multiply the numbers:

3 \cdot 5 = 15

= \frac{5}{15}

= \frac{4 10}{15} + \frac{5}{15}

Apply rule

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

= \frac{4 10 + 5}{15}

= \frac{4 10 + 5}{15}

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Frequently Asked Questions (FAQ)

What is the value of cos(arcsin(1/3)-arctan(1/2)) ?
The value of cos(arcsin(1/3)-arctan(1/2)) is (4sqrt(10)+sqrt(5))/(15)