What is the value of cos(2arctan(3)) ?

The value of cos(2arctan(3)) is -4/5

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cos(2arctan(3))

Solution

cos (2 arctan (3))

Solution

- \frac{4}{5}

Decimal Notation

- 0.8

Solution steps

cos (2 arctan (3))

Rewrite using trig identities:

2 cos^{2} (arctan (3)) - 1

cos (2 arctan (3))

Use the Double Angle identity:

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

= 2 cos^{2} (arctan (3)) - 1

= 2 cos^{2} (arctan (3)) - 1

Rewrite using trig identities:

cos (arctan (3)) = \frac{10}{10}

cos (arctan (3))

Rewrite using trig identities:

cos (arctan (3)) = \frac{1 + 3 ^{2}}{1 + 3 ^{2}}

Use the following identity:

cos (arctan (x)) = \frac{1 + x ^{2}}{1 + x ^{2}}

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

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

Simplify

= \frac{10}{10}

= 2 (\frac{10}{10})^{2} - 1

Simplify

2 (\frac{10}{10})^{2} - 1 : - \frac{4}{5}

2 (\frac{10}{10})^{2} - 1

2 (\frac{10}{10})^{2} = \frac{1}{5}

2 (\frac{10}{10})^{2}

(\frac{10}{10})^{2} = \frac{1}{10}

(\frac{10}{10})^{2}

Apply exponent rule:

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

= \frac{( 10 ) ^{2}}{1 0 ^{2}}

(10)^{2} : 10

Apply radical rule:

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

= (1 0^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 1 0^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 10

= \frac{10}{1 0 ^{2}}

Cancel the common factor:

10

= \frac{1}{10}

= 2 \cdot \frac{1}{10}

Multiply fractions:

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

= \frac{1 \cdot 2}{10}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{10}

Cancel the common factor:

2

= \frac{1}{5}

= \frac{1}{5} - 1

Convert element to fraction:

1 = \frac{1 \cdot 5}{5}

= - \frac{1 \cdot 5}{5} + \frac{1}{5}

Since the denominators are equal, combine the fractions:

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

= \frac{- 1 \cdot 5 + 1}{5}

- 1 \cdot 5 + 1 = - 4

- 1 \cdot 5 + 1

Multiply the numbers:

1 \cdot 5 = 5

= - 5 + 1

Add/Subtract the numbers:

- 5 + 1 = - 4

= - 4

= \frac{- 4}{5}

Apply the fraction rule:

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

= - \frac{4}{5}

= - \frac{4}{5}

Popular Examples

sin(43.5)

sin (43. 5^{\circ})

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csc (- \frac{7 π}{3})

csc(6pi)

csc (6 π)

tan((25pi)/4)

tan (\frac{25 π}{4})

Frequently Asked Questions (FAQ)

What is the value of cos(2arctan(3)) ?
The value of cos(2arctan(3)) is -4/5