What is the value of cos(arctan(2/3)-arccos(3/5)) ?

The value of cos(arctan(2/3)-arccos(3/5)) is (17sqrt(13))/(65)

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cos(arctan(2/3)-arccos(3/5))

Solution

cos (arctan (\frac{2}{3}) - arccos (\frac{3}{5}))

Solution

\frac{17 13}{65}

Decimal Notation

0.94299 \dots

Solution steps

cos (arctan (\frac{2}{3}) - arccos (\frac{3}{5}))

Rewrite using trig identities:

cos (arctan (\frac{2}{3})) cos (arccos (\frac{3}{5})) + sin (arctan (\frac{2}{3})) sin (arccos (\frac{3}{5}))

cos (arctan (\frac{2}{3}) - arccos (\frac{3}{5}))

Use the Angle Difference identity:

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

= cos (arctan (\frac{2}{3})) cos (arccos (\frac{3}{5})) + sin (arctan (\frac{2}{3})) sin (arccos (\frac{3}{5}))

= cos (arctan (\frac{2}{3})) cos (arccos (\frac{3}{5})) + sin (arctan (\frac{2}{3})) sin (arccos (\frac{3}{5}))

Rewrite using trig identities:

cos (arctan (\frac{2}{3})) = \frac{3 13}{13}

cos (arctan (\frac{2}{3}))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{3 13}{13}

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{2}{3})) = \frac{2 13}{13}

sin (arctan (\frac{2}{3}))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{2 13}{13}

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{3 13}{13} \cdot \frac{3}{5} + \frac{2 13}{13} \cdot \frac{4}{5}

Simplify

\frac{3 13}{13} \cdot \frac{3}{5} + \frac{2 13}{13} \cdot \frac{4}{5} : \frac{17 13}{65}

\frac{3 13}{13} \cdot \frac{3}{5} + \frac{2 13}{13} \cdot \frac{4}{5}

\frac{3 13}{13} \cdot \frac{3}{5} = \frac{9 13}{65}

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

Multiply fractions:

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

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

Multiply the numbers:

3 \cdot 3 = 9

= \frac{9 13}{13 \cdot 5}

Multiply the numbers:

13 \cdot 5 = 65

= \frac{9 13}{65}

\frac{2 13}{13} \cdot \frac{4}{5} = \frac{8 13}{65}

\frac{2 13}{13} \cdot \frac{4}{5}

Multiply fractions:

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

= \frac{2 13 \cdot 4}{13 \cdot 5}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{8 13}{13 \cdot 5}

Multiply the numbers:

13 \cdot 5 = 65

= \frac{8 13}{65}

= \frac{9 13}{65} + \frac{8 13}{65}

Apply rule

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

= \frac{9 13 + 8 13}{65}

Add similar elements:

913 + 813 = 1713

= \frac{17 13}{65}

= \frac{17 13}{65}

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Frequently Asked Questions (FAQ)

What is the value of cos(arctan(2/3)-arccos(3/5)) ?
The value of cos(arctan(2/3)-arccos(3/5)) is (17sqrt(13))/(65)