What is the value of cos(arctan(8/15)-arccos(3/5)) ?

The value of cos(arctan(8/15)-arccos(3/5)) is 77/85

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cos(arctan(8/15)-arccos(3/5))

Solution

cos (arctan (\frac{8}{15}) - arccos (\frac{3}{5}))

Solution

\frac{77}{85}

Decimal Notation

0.90588 \dots

Solution steps

cos (arctan (\frac{8}{15}) - arccos (\frac{3}{5}))

Rewrite using trig identities:

cos (arctan (\frac{8}{15})) cos (arccos (\frac{3}{5})) + sin (arctan (\frac{8}{15})) sin (arccos (\frac{3}{5}))

cos (arctan (\frac{8}{15}) - arccos (\frac{3}{5}))

Use the Angle Difference identity:

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

= cos (arctan (\frac{8}{15})) cos (arccos (\frac{3}{5})) + sin (arctan (\frac{8}{15})) sin (arccos (\frac{3}{5}))

= cos (arctan (\frac{8}{15})) cos (arccos (\frac{3}{5})) + sin (arctan (\frac{8}{15})) sin (arccos (\frac{3}{5}))

Rewrite using trig identities:

cos (arctan (\frac{8}{15})) = \frac{15}{17}

cos (arctan (\frac{8}{15}))

Rewrite using trig identities:

cos (arctan (\frac{8}{15})) = \frac{1 + ( \frac{8}{15} ) ^{2}}{1 + ( \frac{8}{15} ) ^{2}}

Use the following identity:

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

= \frac{1 + ( \frac{8}{15} ) ^{2}}{1 + ( \frac{8}{15} ) ^{2}}

= \frac{1 + ( \frac{8}{15} ) ^{2}}{1 + ( \frac{8}{15} ) ^{2}}

Simplify

= \frac{15}{17}

Rewrite using trig identities:

cos (arccos (\frac{3}{5})) = \frac{3}{5}

Use the following identity:

cos (arccos (x)) = x

= \frac{3}{5}

Rewrite using trig identities:

sin (arctan (\frac{8}{15})) = \frac{8}{17}

sin (arctan (\frac{8}{15}))

Rewrite using trig identities:

sin (arctan (\frac{8}{15})) = \frac{( \frac{8}{15} ) 1 + ( \frac{8}{15} ) ^{2}}{1 + ( \frac{8}{15} ) ^{2}}

Use the following identity:

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

= \frac{( \frac{8}{15} ) 1 + ( \frac{8}{15} ) ^{2}}{1 + ( \frac{8}{15} ) ^{2}}

= \frac{\frac{8}{15} 1 + ( \frac{8}{15} ) ^{2}}{1 + ( \frac{8}{15} ) ^{2}}

Simplify

= \frac{8}{17}

Rewrite using trig identities:

sin (arccos (\frac{3}{5})) = \frac{4}{5}

sin (arccos (\frac{3}{5}))

Rewrite using trig identities:

sin (arccos (\frac{3}{5})) = 1 - (\frac{3}{5})^{2}

Use the following identity:

sin (arccos (x)) = 1 - x^{2}

= 1 - (\frac{3}{5})^{2}

= 1 - (\frac{3}{5})^{2}

Simplify

= \frac{4}{5}

= \frac{15}{17} \cdot \frac{3}{5} + \frac{8}{17} \cdot \frac{4}{5}

Simplify

\frac{15}{17} \cdot \frac{3}{5} + \frac{8}{17} \cdot \frac{4}{5} : \frac{77}{85}

\frac{15}{17} \cdot \frac{3}{5} + \frac{8}{17} \cdot \frac{4}{5}

\frac{15}{17} \cdot \frac{3}{5} = \frac{9}{17}

\frac{15}{17} \cdot \frac{3}{5}

Cross-cancel common factor:

5

= \frac{3}{17} \cdot \frac{3}{1}

Multiply fractions:

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

= \frac{3 \cdot 3}{17 \cdot 1}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{9}{17 \cdot 1}

Multiply the numbers:

17 \cdot 1 = 17

= \frac{9}{17}

\frac{8}{17} \cdot \frac{4}{5} = \frac{32}{85}

\frac{8}{17} \cdot \frac{4}{5}

Multiply fractions:

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

= \frac{8 \cdot 4}{17 \cdot 5}

Multiply the numbers:

8 \cdot 4 = 32

= \frac{32}{17 \cdot 5}

Multiply the numbers:

17 \cdot 5 = 85

= \frac{32}{85}

= \frac{9}{17} + \frac{32}{85}

Least Common Multiplier of

17, 85 : 85

17, 85

Least Common Multiplier (LCM)

Prime factorization of

17 : 17

17

17

is a prime number, therefore no factorization is possible

= 17

Prime factorization of

85 : 5 \cdot 17

85

85

divides by

585 = 17 \cdot 5

= 5 \cdot 17

5, 17

are all prime numbers, therefore no further factorization is possible

= 5 \cdot 17

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

17

85

= 17 \cdot 5

Multiply the numbers:

17 \cdot 5 = 85

= 85

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

85

For

\frac{9}{17} :

multiply the denominator and numerator by

5

\frac{9}{17} = \frac{9 \cdot 5}{17 \cdot 5} = \frac{45}{85}

= \frac{45}{85} + \frac{32}{85}

Since the denominators are equal, combine the fractions:

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

= \frac{45 + 32}{85}

Add the numbers:

45 + 32 = 77

= \frac{77}{85}

= \frac{77}{85}

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Frequently Asked Questions (FAQ)

What is the value of cos(arctan(8/15)-arccos(3/5)) ?
The value of cos(arctan(8/15)-arccos(3/5)) is 77/85