What is the value of cos(arcsin(1/(sqrt(5)))+arctan(1/3)) ?

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

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cos(arcsin(1/(sqrt(5)))+arctan(1/3))

Solution

cos (arcsin (\frac{1}{5}) + arctan (\frac{1}{3}))

Solution

\frac{2}{2}

Decimal Notation

0.70710 \dots

Solution steps

cos (arcsin (\frac{1}{5}) + arctan (\frac{1}{3}))

cos (arcsin (\frac{1}{5}) + arctan (\frac{1}{3})) = cos (arcsin (\frac{5}{5}) + arctan (\frac{1}{3}))

cos (arcsin (\frac{1}{5}) + arctan (\frac{1}{3}))

\frac{1}{5} = \frac{5}{5}

\frac{1}{5}

Multiply by the conjugate

\frac{5}{5}

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

1 \cdot 5 = 5

55 = 5

55

Apply radical rule:

a a = a

55 = 5

= 5

= \frac{5}{5}

= cos (arcsin (\frac{5}{5}) + arctan (\frac{1}{3}))

= cos (arcsin (\frac{5}{5}) + arctan (\frac{1}{3}))

Rewrite using trig identities:

cos (arcsin (\frac{5}{5})) cos (arctan (\frac{1}{3})) - sin (arcsin (\frac{5}{5})) sin (arctan (\frac{1}{3}))

cos (arcsin (\frac{5}{5}) + arctan (\frac{1}{3}))

Use the Angle Sum identity:

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

= cos (arcsin (\frac{5}{5})) cos (arctan (\frac{1}{3})) - sin (arcsin (\frac{5}{5})) sin (arctan (\frac{1}{3}))

= cos (arcsin (\frac{5}{5})) cos (arctan (\frac{1}{3})) - sin (arcsin (\frac{5}{5})) sin (arctan (\frac{1}{3}))

Rewrite using trig identities:

cos (arcsin (\frac{5}{5})) = \frac{2 5}{5}

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{2 5}{5}

Rewrite using trig identities:

cos (arctan (\frac{1}{3})) = \frac{3 10}{10}

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{3 10}{10}

Rewrite using trig identities:

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

Use the following identity:

sin (arcsin (x)) = x

= \frac{5}{5}

Rewrite using trig identities:

sin (arctan (\frac{1}{3})) = \frac{10}{10}

sin (arctan (\frac{1}{3}))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{10}{10}

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

Simplify

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

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

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

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

Multiply fractions:

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

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

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6 5 10}{5 \cdot 10}

Multiply the numbers:

5 \cdot 10 = 50

= \frac{6 5 10}{50}

Cancel the common factor:

2

= \frac{3 5 10}{25}

Simplify

3510 : 3 \cdot 52

3510

Factor integer

10 = 5 \cdot 2

= 35 5 \cdot 2

Apply radical rule:

5 \cdot 2 = 52

= 3552

Apply radical rule:

a a = a

55 = 5

= 3 \cdot 52

= \frac{3 \cdot 5 2}{25}

Multiply the numbers:

3 \cdot 5 = 15

= \frac{15 2}{25}

Cancel the common factor:

5

= \frac{3 2}{5}

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

\frac{5}{5} \cdot \frac{10}{10}

Multiply fractions:

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

= \frac{5 10}{5 \cdot 10}

Multiply the numbers:

5 \cdot 10 = 50

= \frac{5 10}{50}

Simplify

510 : 52

510

Factor integer

10 = 5 \cdot 2

= 5 5 \cdot 2

Apply radical rule:

5 \cdot 2 = 52

= 552

Apply radical rule:

a a = a

55 = 5

= 52

= \frac{5 2}{50}

Cancel the common factor:

5

= \frac{2}{10}

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

Least Common Multiplier of

5, 10 : 10

5, 10

Least Common Multiplier (LCM)

Prime factorization of

5 : 5

5

5

is a prime number, therefore no factorization is possible

= 5

Prime factorization of

10 : 2 \cdot 5

10

10

divides by

210 = 5 \cdot 2

= 2 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 5

Multiply each factor the greatest number of times it occurs in either

5

10

= 5 \cdot 2

Multiply the numbers:

5 \cdot 2 = 10

= 10

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

10

For

\frac{3 2}{5} :

multiply the denominator and numerator by

2

\frac{3 2}{5} = \frac{3 2 \cdot 2}{5 \cdot 2} = \frac{6 2}{10}

= \frac{6 2}{10} - \frac{2}{10}

Since the denominators are equal, combine the fractions:

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

= \frac{6 2 - 2}{10}

Add similar elements:

62 - 2 = 52

= \frac{5 2}{10}

Cancel the common factor:

5

= \frac{2}{2}

= \frac{2}{2}

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Frequently Asked Questions (FAQ)

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