What is the value of sin(pi/3-arctan(1/(sqrt(3)))) ?

The value of sin(pi/3-arctan(1/(sqrt(3)))) is 1/2

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sin(pi/3-arctan(1/(sqrt(3))))

Solution

sin (\frac{π}{3} - arctan (\frac{1}{3}))

Solution

\frac{1}{2}

Decimal Notation

0.5

Solution steps

sin (\frac{π}{3} - arctan (\frac{1}{3}))

Rewrite using trig identities:

sin (\frac{π}{3}) cos (arctan (\frac{3}{3})) - cos (\frac{π}{3}) sin (arctan (\frac{3}{3}))

sin (\frac{π}{3} - arctan (\frac{1}{3}))

Use the Angle Difference identity:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= sin (\frac{π}{3}) cos (arctan (\frac{1}{3})) - cos (\frac{π}{3}) sin (arctan (\frac{1}{3}))

sin (\frac{π}{3}) cos (arctan (\frac{1}{3})) - cos (\frac{π}{3}) sin (arctan (\frac{1}{3})) = sin (\frac{π}{3}) cos (arctan (\frac{3}{3})) - cos (\frac{π}{3}) sin (arctan (\frac{3}{3}))

sin (\frac{π}{3}) cos (arctan (\frac{1}{3})) - cos (\frac{π}{3}) sin (arctan (\frac{1}{3}))

\frac{1}{3} = \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

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

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

= sin (\frac{π}{3}) cos (arctan (\frac{3}{3})) - cos (\frac{π}{3}) sin (arctan (\frac{1}{3}))

\frac{1}{3} = \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

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

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

= sin (\frac{π}{3}) cos (arctan (\frac{3}{3})) - cos (\frac{π}{3}) sin (arctan (\frac{3}{3}))

= sin (\frac{π}{3}) cos (arctan (\frac{3}{3})) - cos (\frac{π}{3}) sin (arctan (\frac{3}{3}))

= sin (\frac{π}{3}) cos (arctan (\frac{3}{3})) - cos (\frac{π}{3}) sin (arctan (\frac{3}{3}))

Use the following trivial identity:

sin (\frac{π}{3}) = \frac{3}{2}

sin (\frac{π}{3})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

Rewrite using trig identities:

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

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{3}{2}

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

cos (\frac{π}{3})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{1}{2}

Rewrite using trig identities:

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

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{1}{2}

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

Simplify

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

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

\frac{3}{2} \cdot \frac{3}{2} = \frac{3}{4}

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

Multiply fractions:

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

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

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3}{4}

\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

= \frac{3}{4} - \frac{1}{4}

Apply rule

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

= \frac{3 - 1}{4}

Subtract the numbers:

3 - 1 = 2

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

= \frac{1}{2}

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Frequently Asked Questions (FAQ)

What is the value of sin(pi/3-arctan(1/(sqrt(3)))) ?
The value of sin(pi/3-arctan(1/(sqrt(3)))) is 1/2