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

The value of cos(2arcsin(5/13)) is 119/169

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cos(2arcsin(5/13))

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

\frac{119}{169}

Decimal Notation

0.70414 \dots

Solution steps

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

Rewrite using trig identities:

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

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

Use the Double Angle identity:

cos (2 x) = 1 - 2 sin^{2} (x)

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

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

Rewrite using trig identities:

sin (arcsin (\frac{5}{13})) = \frac{5}{13}

Use the following identity:

sin (arcsin (x)) = x

= \frac{5}{13}

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

Simplify

1 - 2 (\frac{5}{13})^{2} : \frac{119}{169}

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

2 (\frac{5}{13})^{2} = \frac{50}{169}

2 (\frac{5}{13})^{2}

(\frac{5}{13})^{2} = \frac{5 ^{2}}{1 3 ^{2}}

(\frac{5}{13})^{2}

Apply exponent rule:

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

= \frac{5 ^{2}}{1 3 ^{2}}

= 2 \cdot \frac{5 ^{2}}{1 3 ^{2}}

Multiply fractions:

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

= \frac{5 ^{2} \cdot 2}{1 3 ^{2}}

5^{2} \cdot 2 = 50

5^{2} \cdot 2

5^{2} = 25

= 25 \cdot 2

Multiply the numbers:

25 \cdot 2 = 50

= 50

= \frac{50}{1 3 ^{2}}

1 3^{2} = 169

= \frac{50}{169}

= 1 - \frac{50}{169}

Convert element to fraction:

1 = \frac{1 \cdot 169}{169}

= \frac{1 \cdot 169}{169} - \frac{50}{169}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 169 - 50}{169}

1 \cdot 169 - 50 = 119

1 \cdot 169 - 50

Multiply the numbers:

1 \cdot 169 = 169

= 169 - 50

Subtract the numbers:

169 - 50 = 119

= 119

= \frac{119}{169}

= \frac{119}{169}

sec (33 0^{\circ})

sec (22 5^{\circ})

arctan (\frac{6}{5})

sin (arccos (\frac{1}{2}))

cos (arcsin (\frac{3}{2}))

What is the value of cos(2arcsin(5/13)) ?
The value of cos(2arcsin(5/13)) is 119/169