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))))

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

\frac{1}{2}

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}) 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}

Rewrite using trig identities:

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

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

Rewrite using trig identities:

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

5 + 4 cos (\frac{π}{3})

ln (tan (\frac{π}{4}))

\frac{cos ( \frac{π}{6} )}{( 1 - sin ( \frac{π}{6} ) ) ^{2}}

arctan (\frac{5}{15})

cos (12 9^{\circ})

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