What is the value of cos(arctan(sqrt(3))+arcsin(1/3)) ?

The value of cos(arctan(sqrt(3))+arcsin(1/3)) is (sqrt(2))/3-(sqrt(3))/6

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cos(arctan(sqrt(3))+arcsin(1/3))

Solution

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

Solution

\frac{2}{3} - \frac{3}{6}

Decimal Notation

0.18272 \dots

Solution steps

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

Rewrite using trig identities:

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

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

Use the Angle Sum identity:

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

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

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

Rewrite using trig identities:

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

cos (arctan (3))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \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:

sin (arctan (3)) = \frac{3}{2}

sin (arctan (3))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{3}{2}

Rewrite using trig identities:

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

Use the following identity:

sin (arcsin (x)) = x

= \frac{1}{3}

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

Simplify

\frac{1}{2} \cdot \frac{2 2}{3} - \frac{3}{2} \cdot \frac{1}{3} : \frac{2}{3} - \frac{3}{6}

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

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

Multiply:

1 \cdot 2 = 2

= \frac{2}{3}

\frac{3}{2} \cdot \frac{1}{3} = \frac{3}{6}

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

Multiply fractions:

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

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

Multiply:

3 \cdot 1 = 3

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

Multiply the numbers:

2 \cdot 3 = 6

= \frac{3}{6}

= \frac{2}{3} - \frac{3}{6}

= \frac{2}{3} - \frac{3}{6}

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Frequently Asked Questions (FAQ)

What is the value of cos(arctan(sqrt(3))+arcsin(1/3)) ?
The value of cos(arctan(sqrt(3))+arcsin(1/3)) is (sqrt(2))/3-(sqrt(3))/6