What is the value of sin(2arctan(sqrt(3))) ?

The value of sin(2arctan(sqrt(3))) is (sqrt(3))/2

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sin(2arctan(sqrt(3)))

sin (2 arctan (3))

\frac{3}{2}

Decimal Notation

0.86602 \dots

Solution steps

sin (2 arctan (3))

Rewrite using trig identities:

2 sin (arctan (3)) cos (arctan (3))

sin (2 arctan (3))

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= 2 sin (arctan (3)) cos (arctan (3))

= 2 sin (arctan (3)) cos (arctan (3))

Rewrite using trig identities:

sin (arctan (3)) = \frac{3}{2}

sin (arctan (3))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{3}{2}

Rewrite using trig identities:

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

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{1}{2}

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

Simplify

2 \cdot \frac{3}{2} \cdot \frac{1}{2} : \frac{3}{2}

2 \cdot \frac{3}{2} \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

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

Multiply:

3 \cdot 1 = 3

= \frac{3}{2}

= \frac{3}{2}

4 sin (2 \cdot \frac{π}{2})

sin (\frac{180}{π})

2.5 tan (4 0^{\circ})

1 5^{2} + 4 0^{2} - 2 (15) (40) cos (6 0^{\circ})

cos (0.1) - \frac{199}{200}

What is the value of sin(2arctan(sqrt(3))) ?
The value of sin(2arctan(sqrt(3))) is (sqrt(3))/2