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

The value of cos(arcsin(4/5)-arctan(12/5)) is 63/65

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

Solution

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

Solution

\frac{63}{65}

Decimal Notation

0.96923 \dots

Solution steps

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

Rewrite using trig identities:

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

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

Use the Angle Difference identity:

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

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

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

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:

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

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Rewrite using trig identities:

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

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{12}{13}

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

Simplify

\frac{3}{5} \cdot \frac{5}{13} + \frac{4}{5} \cdot \frac{12}{13} : \frac{63}{65}

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

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

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

Cross-cancel:

5

= \frac{3}{13}

\frac{4}{5} \cdot \frac{12}{13} = \frac{48}{65}

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

Multiply fractions:

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

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

Multiply the numbers:

4 \cdot 12 = 48

= \frac{48}{5 \cdot 13}

Multiply the numbers:

5 \cdot 13 = 65

= \frac{48}{65}

= \frac{3}{13} + \frac{48}{65}

Least Common Multiplier of

13, 65 : 65

13, 65

Least Common Multiplier (LCM)

Prime factorization of

13 : 13

13

13

is a prime number, therefore no factorization is possible

= 13

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

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

13

65

= 13 \cdot 5

Multiply the numbers:

13 \cdot 5 = 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{3}{13} :

multiply the denominator and numerator by

5

\frac{3}{13} = \frac{3 \cdot 5}{13 \cdot 5} = \frac{15}{65}

= \frac{15}{65} + \frac{48}{65}

Since the denominators are equal, combine the fractions:

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

= \frac{15 + 48}{65}

Add the numbers:

15 + 48 = 63

= \frac{63}{65}

= \frac{63}{65}

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Frequently Asked Questions (FAQ)

What is the value of cos(arcsin(4/5)-arctan(12/5)) ?
The value of cos(arcsin(4/5)-arctan(12/5)) is 63/65