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

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

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Popular Trigonometry >

cos(2arctan(-2))

Solution

cos (2 arctan (- 2))

Solution

- \frac{3}{5}

Decimal

- 0.6

Solution steps

cos (2 arctan (- 2))

Rewrite using trig identities:

1 - 2 sin^{2} (arctan (2))

cos (2 arctan (- 2))

Use the Double Angle identity:

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

= 1 - 2 sin^{2} (arctan (- 2))

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 2) = - arctan (2)

= 1 - 2 sin^{2} (- arctan (2))

Use the following property:

sin (- x) = - sin (x)

sin (- arctan (2)) = - sin (arctan (2))

= 1 - 2 (- sin (arctan (2)))^{2}

Simplify

= 1 - 2 sin^{2} (arctan (2))

= 1 - 2 sin^{2} (arctan (2))

Rewrite using trig identities:

sin (arctan (2)) = \frac{2 5}{5}

sin (arctan (2))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{2 5}{5}

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

Simplify

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

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

2 (\frac{2 5}{5})^{2} = \frac{8}{5}

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

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

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

Apply exponent rule:

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

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

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(25)^{2} = 2^{2} (5)^{2}

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

(5)^{2} : 5

Apply radical rule:

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

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

Apply exponent rule:

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

= 5^{\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

= 5

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

Cancel the common factor:

5

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

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

Multiply fractions:

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

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

2^{2} \cdot 2 = 2^{3}

2^{2} \cdot 2

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

2^{2} \cdot 2 = 2^{2 + 1}

= 2^{2 + 1}

Add the numbers:

2 + 1 = 3

= 2^{3}

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

2^{3} = 8

= \frac{8}{5}

= 1 - \frac{8}{5}

Convert element to fraction:

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

= \frac{1 \cdot 5}{5} - \frac{8}{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 - 8}{5}

1 \cdot 5 - 8 = - 3

1 \cdot 5 - 8

Multiply the numbers:

1 \cdot 5 = 5

= 5 - 8

Subtract the numbers:

5 - 8 = - 3

= - 3

= \frac{- 3}{5}

Apply the fraction rule:

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

= - \frac{3}{5}

= - \frac{3}{5}

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Frequently Asked Questions (FAQ)

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