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

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

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

Apply the fraction rule:

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

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

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

Apply the fraction rule:

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

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

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

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

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

Simplify

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

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

Remove parentheses:

(- a) = - a

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

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

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

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

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

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

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

Apply the fraction rule:

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

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

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

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

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

Apply the fraction rule:

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

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

Join

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

\frac{16}{9} - 1

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

16 - 1 \cdot 9 = 7

16 - 1 \cdot 9

Multiply the numbers:

1 \cdot 9 = 9

= 16 - 9

Subtract the numbers:

16 - 9 = 7

= 7

= \frac{7}{9}

= \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}}

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}

Popular Examples

arcsin(9/11)

arcsin (\frac{9}{11})

tan(sqrt(2))

tan (2)

cot(-sqrt(3))

cot (- 3)

arccos(3/10)

arccos (\frac{3}{10})

arctan^2(1)

arctan^{2} (1)

Frequently Asked Questions (FAQ)

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