What is the value of cos(arcsin(4/5)+arctan(5/12)) ?

The value of cos(arcsin(4/5)+arctan(5/12)) is 16/65

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cos(arcsin(4/5)+arctan(5/12))

Solution

cos (arcsin (\frac{4}{5}) + arctan (\frac{5}{12}))

Solution

\frac{16}{65}

Decimal Notation

0.24615 \dots

Solution steps

cos (arcsin (\frac{4}{5}) + arctan (\frac{5}{12}))

Rewrite using trig identities:

cos (arctan (\frac{5}{12})) cos (arcsin (\frac{4}{5})) - sin (arctan (\frac{5}{12})) sin (arcsin (\frac{4}{5}))

cos (arctan (\frac{5}{12}) + arcsin (\frac{4}{5}))

Use the Angle Sum identity:

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

= cos (arctan (\frac{5}{12})) cos (arcsin (\frac{4}{5})) - sin (arctan (\frac{5}{12})) sin (arcsin (\frac{4}{5}))

= cos (arctan (\frac{5}{12})) cos (arcsin (\frac{4}{5})) - sin (arctan (\frac{5}{12})) sin (arcsin (\frac{4}{5}))

Rewrite using trig identities:

cos (arctan (\frac{5}{12})) = \frac{12}{13}

cos (arctan (\frac{5}{12}))

Rewrite using trig identities:

cos (arctan (\frac{5}{12})) = \frac{1 + ( \frac{5}{12} ) ^{2}}{1 + ( \frac{5}{12} ) ^{2}}

Use the following identity:

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

= \frac{1 + ( \frac{5}{12} ) ^{2}}{1 + ( \frac{5}{12} ) ^{2}}

= \frac{1 + ( \frac{5}{12} ) ^{2}}{1 + ( \frac{5}{12} ) ^{2}}

Simplify

= \frac{12}{13}

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:

sin (arctan (\frac{5}{12})) = \frac{5}{13}

sin (arctan (\frac{5}{12}))

Rewrite using trig identities:

sin (arctan (\frac{5}{12})) = \frac{( \frac{5}{12} ) 1 + ( \frac{5}{12} ) ^{2}}{1 + ( \frac{5}{12} ) ^{2}}

Use the following identity:

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

= \frac{( \frac{5}{12} ) 1 + ( \frac{5}{12} ) ^{2}}{1 + ( \frac{5}{12} ) ^{2}}

= \frac{\frac{5}{12} 1 + ( \frac{5}{12} ) ^{2}}{1 + ( \frac{5}{12} ) ^{2}}

Simplify

= \frac{5}{13}

Rewrite using trig identities:

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

Use the following identity:

sin (arcsin (x)) = x

= \frac{4}{5}

= \frac{12}{13} \cdot \frac{3}{5} - \frac{5}{13} \cdot \frac{4}{5}

Simplify

\frac{12}{13} \cdot \frac{3}{5} - \frac{5}{13} \cdot \frac{4}{5} : \frac{16}{65}

\frac{12}{13} \cdot \frac{3}{5} - \frac{5}{13} \cdot \frac{4}{5}

\frac{12}{13} \cdot \frac{3}{5} = \frac{36}{65}

\frac{12}{13} \cdot \frac{3}{5}

Multiply fractions:

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

= \frac{12 \cdot 3}{13 \cdot 5}

Multiply the numbers:

12 \cdot 3 = 36

= \frac{36}{13 \cdot 5}

Multiply the numbers:

13 \cdot 5 = 65

= \frac{36}{65}

\frac{5}{13} \cdot \frac{4}{5} = \frac{4}{13}

\frac{5}{13} \cdot \frac{4}{5}

Cross-cancel:

5

= \frac{4}{13}

= \frac{36}{65} - \frac{4}{13}

Least Common Multiplier of

65, 13 : 65

65, 13

Least Common Multiplier (LCM)

Prime factorization of

65 : 5 \cdot 13

65

65

divides by

565 = 13 \cdot 5

= 5 \cdot 13

5, 13

are all prime numbers, therefore no further factorization is possible

= 5 \cdot 13

Prime factorization of

13 : 13

13

13

is a prime number, therefore no factorization is possible

= 13

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

65

13

= 5 \cdot 13

Multiply the numbers:

5 \cdot 13 = 65

= 65

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

65

For

\frac{4}{13} :

multiply the denominator and numerator by

5

\frac{4}{13} = \frac{4 \cdot 5}{13 \cdot 5} = \frac{20}{65}

= \frac{36}{65} - \frac{20}{65}

Since the denominators are equal, combine the fractions:

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

= \frac{36 - 20}{65}

Subtract the numbers:

36 - 20 = 16

= \frac{16}{65}

= \frac{16}{65}

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Frequently Asked Questions (FAQ)

What is the value of cos(arcsin(4/5)+arctan(5/12)) ?
The value of cos(arcsin(4/5)+arctan(5/12)) is 16/65