What is the value of tan(2arccos(3/5)) ?

The value of tan(2arccos(3/5)) is -24/7

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tan(2arccos(3/5))

Solution

tan (2 arccos (\frac{3}{5}))

Solution

- \frac{24}{7}

Decimal Notation

- 3.42857 \dots

Solution steps

tan (2 arccos (\frac{3}{5}))

Rewrite using trig identities:

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

tan (2 arccos (\frac{3}{5}))

Use the Double Angle identity:

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

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

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

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

Simplify

\frac{2 \cdot \frac{4}{3}}{1 - ( \frac{4}{3} ) ^{2}} : - \frac{24}{7}

\frac{2 \cdot \frac{4}{3}}{1 - ( \frac{4}{3} ) ^{2}}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{2 \cdot \frac{4}{3}}{1 - \frac{4 ^{2}}{3 ^{2}}}

Multiply

2 \cdot \frac{4}{3} : \frac{8}{3}

2 \cdot \frac{4}{3}

Multiply fractions:

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

= \frac{4 \cdot 2}{3}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{8}{3}

= \frac{\frac{8}{3}}{1 - \frac{4 ^{2}}{3 ^{2}}}

\frac{4 ^{2}}{3 ^{2}} = \frac{16}{9}

\frac{4 ^{2}}{3 ^{2}}

4^{2} = 16

= \frac{16}{3 ^{2}}

3^{2} = 9

= \frac{16}{9}

= \frac{\frac{8}{3}}{1 - \frac{16}{9}}

Apply the fraction rule:

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

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

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(2arccos(3/5)) ?
The value of tan(2arccos(3/5)) is -24/7