What is the value of tan(2arccos(-4/5)) ?

The value of tan(2arccos(-4/5)) is -24/7

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tan(2arccos(-4/5))

Solution

tan (2 arccos (- \frac{4}{5}))

Solution

- \frac{24}{7}

Decimal Notation

- 3.42857 \dots

Solution steps

tan (2 arccos (- \frac{4}{5}))

Rewrite using trig identities:

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

tan (2 arccos (- \frac{4}{5}))

Use the Double Angle identity:

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

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

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

Rewrite using trig identities:

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

tan (arccos (- \frac{4}{5}))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= - \frac{3}{4}

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

Simplify

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

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

Remove parentheses:

(- a) = - a

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

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

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

Apply exponent rule:

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

n

is even

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

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

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

Apply the fraction rule:

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

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

Apply exponent rule:

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

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

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

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

3^{2} = 9

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

4^{2} = 16

= \frac{9}{16}

= - \frac{2 \cdot \frac{3}{4}}{- \frac{9}{16} + 1}

Join

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

1 - \frac{9}{16}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

1 \cdot 16 - 9 = 7

1 \cdot 16 - 9

Multiply the numbers:

1 \cdot 16 = 16

= 16 - 9

Subtract the numbers:

16 - 9 = 7

= 7

= \frac{7}{16}

= - \frac{2 \cdot \frac{3}{4}}{\frac{7}{16}}

Simplify

\frac{2 \cdot \frac{3}{4}}{\frac{7}{16}} : \frac{32 \cdot \frac{3}{4}}{7}

\frac{2 \cdot \frac{3}{4}}{\frac{7}{16}}

Apply the fraction rule:

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

= \frac{2 \cdot \frac{3}{4} \cdot 16}{7}

Multiply the numbers:

2 \cdot 16 = 32

= \frac{32 \cdot \frac{3}{4}}{7}

= - \frac{32 \cdot \frac{3}{4}}{7}

Multiply

32 \cdot \frac{3}{4} : 24

32 \cdot \frac{3}{4}

Multiply fractions:

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

= \frac{3 \cdot 32}{4}

Multiply the numbers:

3 \cdot 32 = 96

= \frac{96}{4}

Divide the numbers:

\frac{96}{4} = 24

= 24

= - \frac{24}{7}

= - \frac{24}{7}

Popular Examples

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Frequently Asked Questions (FAQ)

What is the value of tan(2arccos(-4/5)) ?
The value of tan(2arccos(-4/5)) is -24/7