What is the value of cos(arctan(3/4)+arccos(8/17)) ?

The value of cos(arctan(3/4)+arccos(8/17)) is -13/85

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Popular Trigonometry >

cos(arctan(3/4)+arccos(8/17))

Solution

cos (arctan (\frac{3}{4}) + arccos (\frac{8}{17}))

Solution

- \frac{13}{85}

Decimal

- 0.15294 \dots

Solution steps

cos (arctan (\frac{3}{4}) + arccos (\frac{8}{17}))

Rewrite using trig identities:

cos (arctan (\frac{3}{4})) cos (arccos (\frac{8}{17})) - sin (arctan (\frac{3}{4})) sin (arccos (\frac{8}{17}))

cos (arctan (\frac{3}{4}) + arccos (\frac{8}{17}))

Use the Angle Sum identity:

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

= cos (arctan (\frac{3}{4})) cos (arccos (\frac{8}{17})) - sin (arctan (\frac{3}{4})) sin (arccos (\frac{8}{17}))

= cos (arctan (\frac{3}{4})) cos (arccos (\frac{8}{17})) - sin (arctan (\frac{3}{4})) sin (arccos (\frac{8}{17}))

Rewrite using trig identities:

cos (arctan (\frac{3}{4})) = \frac{4}{5}

cos (arctan (\frac{3}{4}))

Rewrite using trig identities:

cos (arctan (\frac{3}{4})) = \frac{1 + ( \frac{3}{4} ) ^{2}}{1 + ( \frac{3}{4} ) ^{2}}

Use the following identity:

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

= \frac{1 + ( \frac{3}{4} ) ^{2}}{1 + ( \frac{3}{4} ) ^{2}}

= \frac{1 + ( \frac{3}{4} ) ^{2}}{1 + ( \frac{3}{4} ) ^{2}}

Simplify

= \frac{4}{5}

Rewrite using trig identities:

cos (arccos (\frac{8}{17})) = \frac{8}{17}

Use the following identity:

cos (arccos (x)) = x

= \frac{8}{17}

Rewrite using trig identities:

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

sin (arctan (\frac{3}{4}))

Rewrite using trig identities:

sin (arctan (\frac{3}{4})) = \frac{( \frac{3}{4} ) 1 + ( \frac{3}{4} ) ^{2}}{1 + ( \frac{3}{4} ) ^{2}}

Use the following identity:

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

= \frac{( \frac{3}{4} ) 1 + ( \frac{3}{4} ) ^{2}}{1 + ( \frac{3}{4} ) ^{2}}

= \frac{\frac{3}{4} 1 + ( \frac{3}{4} ) ^{2}}{1 + ( \frac{3}{4} ) ^{2}}

Simplify

= \frac{3}{5}

Rewrite using trig identities:

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

sin (arccos (\frac{8}{17}))

Rewrite using trig identities:

sin (arccos (\frac{8}{17})) = 1 - (\frac{8}{17})^{2}

Use the following identity:

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

= 1 - (\frac{8}{17})^{2}

= 1 - (\frac{8}{17})^{2}

Simplify

= \frac{15}{17}

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

Simplify

\frac{4}{5} \cdot \frac{8}{17} - \frac{3}{5} \cdot \frac{15}{17} : - \frac{13}{85}

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

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

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

Multiply fractions:

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

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

Multiply the numbers:

4 \cdot 8 = 32

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

Multiply the numbers:

5 \cdot 17 = 85

= \frac{32}{85}

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

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

Cross-cancel common factor:

5

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

Multiply fractions:

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

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

Multiply the numbers:

3 \cdot 3 = 9

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

Multiply the numbers:

1 \cdot 17 = 17

= \frac{9}{17}

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

Least Common Multiplier of

85, 17 : 85

85, 17

Least Common Multiplier (LCM)

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

Prime factorization of

17 : 17

17

17

is a prime number, therefore no factorization is possible

= 17

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

85

17

= 5 \cdot 17

Multiply the numbers:

5 \cdot 17 = 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{32}{85} - \frac{45}{85}

Since the denominators are equal, combine the fractions:

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

= \frac{32 - 45}{85}

Subtract the numbers:

32 - 45 = - 13

= \frac{- 13}{85}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{13}{85}

= - \frac{13}{85}

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Frequently Asked Questions (FAQ)

What is the value of cos(arctan(3/4)+arccos(8/17)) ?
The value of cos(arctan(3/4)+arccos(8/17)) is -13/85