What is the value of tan(arccos(3/5)+arcsin(4/5)) ?

The value of tan(arccos(3/5)+arcsin(4/5)) is -24/7

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tan(arccos(3/5)+arcsin(4/5))

Solution

tan (arccos (\frac{3}{5}) + arcsin (\frac{4}{5}))

Solution

- \frac{24}{7}

Decimal Notation

- 3.42857 \dots

Solution steps

tan (arccos (\frac{3}{5}) + arcsin (\frac{4}{5}))

Rewrite using trig identities:

\frac{tan ( arccos ( \frac{3}{5} ) ) + tan ( arcsin ( \frac{4}{5} ) )}{1 - tan ( arccos ( \frac{3}{5} ) ) tan ( arcsin ( \frac{4}{5} ) )}

tan (arccos (\frac{3}{5}) + arcsin (\frac{4}{5}))

Use the Angle Sum identity:

tan (s + t) = \frac{tan ( s ) + tan ( t )}{1 - tan ( s ) tan ( t )}

= \frac{tan ( arccos ( \frac{3}{5} ) ) + tan ( arcsin ( \frac{4}{5} ) )}{1 - tan ( arccos ( \frac{3}{5} ) ) tan ( arcsin ( \frac{4}{5} ) )}

= \frac{tan ( arccos ( \frac{3}{5} ) ) + tan ( arcsin ( \frac{4}{5} ) )}{1 - tan ( arccos ( \frac{3}{5} ) ) tan ( arcsin ( \frac{4}{5} ) )}

Rewrite using trig identities:

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

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

Rewrite using trig identities:

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

Use the following identity:

tan (arccos (x)) = \frac{1 - x ^{2}}{x}

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

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

Simplify

= \frac{4}{3}

Rewrite using trig identities:

tan (arcsin (\frac{4}{5})) = \frac{4}{3}

tan (arcsin (\frac{4}{5}))

Rewrite using trig identities:

tan (arcsin (\frac{4}{5})) = \frac{( \frac{4}{5} ) 1 - ( \frac{4}{5} ) ^{2}}{1 - ( \frac{4}{5} ) ^{2}}

Use the following identity:

tan (arcsin (x)) = \frac{x 1 - x ^{2}}{1 - x ^{2}}

= \frac{( \frac{4}{5} ) 1 - ( \frac{4}{5} ) ^{2}}{1 - ( \frac{4}{5} ) ^{2}}

= \frac{\frac{4}{5} 1 - ( \frac{4}{5} ) ^{2}}{1 - ( \frac{4}{5} ) ^{2}}

Simplify

= \frac{4}{3}

= \frac{\frac{4}{3} + \frac{4}{3}}{1 - \frac{4}{3} \cdot \frac{4}{3}}

Simplify

\frac{\frac{4}{3} + \frac{4}{3}}{1 - \frac{4}{3} \cdot \frac{4}{3}} : - \frac{24}{7}

\frac{\frac{4}{3} + \frac{4}{3}}{1 - \frac{4}{3} \cdot \frac{4}{3}}

Combine the fractions

\frac{4}{3} + \frac{4}{3} : \frac{8}{3}

Apply rule

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

= \frac{4 + 4}{3}

Add the numbers:

4 + 4 = 8

= \frac{8}{3}

= \frac{\frac{8}{3}}{1 - \frac{4}{3} \cdot \frac{4}{3}}

Apply the fraction rule:

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

= \frac{8}{3 ( 1 - \frac{4}{3} \cdot \frac{4}{3} )}

\frac{4}{3} \cdot \frac{4}{3} = \frac{16}{9}

\frac{4}{3} \cdot \frac{4}{3}

Multiply fractions:

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

= \frac{4 \cdot 4}{3 \cdot 3}

Multiply the numbers:

4 \cdot 4 = 16

= \frac{16}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{16}{9}

= \frac{8}{3 ( - \frac{16}{9} + 1 )}

Join

1 - \frac{16}{9} : - \frac{7}{9}

1 - \frac{16}{9}

Convert element to fraction:

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

= \frac{1 \cdot 9}{9} - \frac{16}{9}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 9 - 16}{9}

1 \cdot 9 - 16 = - 7

1 \cdot 9 - 16

Multiply the numbers:

1 \cdot 9 = 9

= 9 - 16

Subtract the numbers:

9 - 16 = - 7

= - 7

= \frac{- 7}{9}

Apply the fraction rule:

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

= - \frac{7}{9}

= \frac{8}{3 ( - \frac{7}{9} )}

Remove parentheses:

(- a) = - a

= \frac{8}{- 3 \cdot \frac{7}{9}}

Apply the fraction rule:

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

= - \frac{8}{3 \cdot \frac{7}{9}}

Multiply

3 \cdot \frac{7}{9} : \frac{7}{3}

3 \cdot \frac{7}{9}

Multiply fractions:

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

= \frac{7 \cdot 3}{9}

Multiply the numbers:

7 \cdot 3 = 21

= \frac{21}{9}

Cancel the common factor:

3

= \frac{7}{3}

= - \frac{8}{\frac{7}{3}}

Simplify

\frac{8}{\frac{7}{3}} : \frac{24}{7}

\frac{8}{\frac{7}{3}}

Apply the fraction rule:

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

= \frac{8 \cdot 3}{7}

Multiply the numbers:

8 \cdot 3 = 24

= \frac{24}{7}

= - \frac{24}{7}

= - \frac{24}{7}

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Frequently Asked Questions (FAQ)

What is the value of tan(arccos(3/5)+arcsin(4/5)) ?
The value of tan(arccos(3/5)+arcsin(4/5)) is -24/7