What is the value of tan(2arcsin(5/13)) ?

The value of tan(2arcsin(5/13)) is 120/119

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tan(2arcsin(5/13))

Solution

tan (2 arcsin (\frac{5}{13}))

Solution

\frac{120}{119}

Decimal Notation

1.00840 \dots

Solution steps

tan (2 arcsin (\frac{5}{13}))

Rewrite using trig identities:

\frac{2 tan ( arcsin ( \frac{5}{13} ) )}{1 - tan ^{2} ( arcsin ( \frac{5}{13} ) )}

tan (2 arcsin (\frac{5}{13}))

Use the Double Angle identity:

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

= \frac{2 tan ( arcsin ( \frac{5}{13} ) )}{1 - tan ^{2} ( arcsin ( \frac{5}{13} ) )}

= \frac{2 tan ( arcsin ( \frac{5}{13} ) )}{1 - tan ^{2} ( arcsin ( \frac{5}{13} ) )}

Rewrite using trig identities:

tan (arcsin (\frac{5}{13})) = \frac{5}{12}

tan (arcsin (\frac{5}{13}))

Rewrite using trig identities:

tan (arcsin (\frac{5}{13})) = \frac{( \frac{5}{13} ) 1 - ( \frac{5}{13} ) ^{2}}{1 - ( \frac{5}{13} ) ^{2}}

Use the following identity:

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

= \frac{( \frac{5}{13} ) 1 - ( \frac{5}{13} ) ^{2}}{1 - ( \frac{5}{13} ) ^{2}}

= \frac{\frac{5}{13} 1 - ( \frac{5}{13} ) ^{2}}{1 - ( \frac{5}{13} ) ^{2}}

Simplify

= \frac{5}{12}

= \frac{2 \cdot \frac{5}{12}}{1 - ( \frac{5}{12} ) ^{2}}

Simplify

\frac{2 \cdot \frac{5}{12}}{1 - ( \frac{5}{12} ) ^{2}} : \frac{120}{119}

\frac{2 \cdot \frac{5}{12}}{1 - ( \frac{5}{12} ) ^{2}}

Apply exponent rule:

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

= \frac{2 \cdot \frac{5}{12}}{1 - \frac{5 ^{2}}{1 2 ^{2}}}

Multiply

2 \cdot \frac{5}{12} : \frac{5}{6}

2 \cdot \frac{5}{12}

Multiply fractions:

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

= \frac{5 \cdot 2}{12}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10}{12}

Cancel the common factor:

2

= \frac{5}{6}

= \frac{\frac{5}{6}}{1 - \frac{5 ^{2}}{1 2 ^{2}}}

\frac{5 ^{2}}{1 2 ^{2}} = \frac{25}{144}

\frac{5 ^{2}}{1 2 ^{2}}

5^{2} = 25

= \frac{25}{1 2 ^{2}}

1 2^{2} = 144

= \frac{25}{144}

= \frac{\frac{5}{6}}{1 - \frac{25}{144}}

Apply the fraction rule:

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

= \frac{5}{6 ( 1 - \frac{25}{144} )}

Join

1 - \frac{25}{144} : \frac{119}{144}

1 - \frac{25}{144}

Convert element to fraction:

1 = \frac{1 \cdot 144}{144}

= \frac{1 \cdot 144}{144} - \frac{25}{144}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 144 - 25}{144}

1 \cdot 144 - 25 = 119

1 \cdot 144 - 25

Multiply the numbers:

1 \cdot 144 = 144

= 144 - 25

Subtract the numbers:

144 - 25 = 119

= 119

= \frac{119}{144}

= \frac{5}{6 \cdot \frac{119}{144}}

Multiply

6 \cdot \frac{119}{144} : \frac{119}{24}

6 \cdot \frac{119}{144}

Multiply fractions:

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

= \frac{119 \cdot 6}{144}

Multiply the numbers:

119 \cdot 6 = 714

= \frac{714}{144}

Cancel the common factor:

6

= \frac{119}{24}

= \frac{5}{\frac{119}{24}}

Apply the fraction rule:

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

= \frac{5 \cdot 24}{119}

Multiply the numbers:

5 \cdot 24 = 120

= \frac{120}{119}

= \frac{120}{119}

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Frequently Asked Questions (FAQ)

What is the value of tan(2arcsin(5/13)) ?
The value of tan(2arcsin(5/13)) is 120/119