What is the tangent of (sin(t))/(2+2cos(t)) ?

The tangent of (sin(t))/(2+2cos(t)) is

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tangent of (sin(t))/(2+2cos(t))

tangent of

\frac{sin ( t )}{2 + 2 cos ( t )}

y = \frac{cos ( a _{0} ) ( 2 + 2 cos ( a _{0} ) ) + 2 sin ^{2} ( a _{0} )}{( 2 + 2 cos ( a _{0} ) ) ^{2}} t + \frac{2 ( sin ( a _{0} ) ( cos ( a _{0} ) + 1 ) - a _{0} ( sin ^{2} ( a _{0} ) + cos ( a _{0} ) ( cos ( a _{0} ) + 1 ) ) )}{( 2 + 2 cos ( a _{0} ) ) ^{2}}

Solution steps

Compute the tangent line to the general point

t = a_{0}

Find the tangent point:

(a_{0}, \frac{sin ( a _{0} )}{2 + 2 cos ( a _{0} )})

Find the slope of

y = \frac{sin ( t )}{2 + 2 cos ( t )} : \frac{d y}{d t} = \frac{cos ( t ) ( 2 + 2 cos ( t ) ) + 2 sin ^{2} ( t )}{( 2 + 2 cos ( t ) ) ^{2}}

EN : T i tl e G e n er a l Eq u a t i o n Sl o p e A t P o in t 2 Eq : m = \frac{cos ( a _{0} ) ( 2 + 2 cos ( a _{0} ) ) + 2 sin ^{2} ( a _{0} )}{( 2 + 2 cos ( a _{0} ) ) ^{2}}

Find the line with slope m=

\frac{cos ( a _{0} ) ( 2 + 2 cos ( a _{0} ) ) + 2 sin ^{2} ( a _{0} )}{( 2 + 2 cos ( a _{0} ) ) ^{2}}

and passing through

(a_{0}, \frac{sin ( a _{0} )}{2 + 2 cos ( a _{0} )}) : y = \frac{cos ( a _{0} ) ( 2 + 2 cos ( a _{0} ) ) + 2 sin ^{2} ( a _{0} )}{( 2 + 2 cos ( a _{0} ) ) ^{2}} t + \frac{2 ( sin ( a _{0} ) ( cos ( a _{0} ) + 1 ) - a _{0} ( sin ^{2} ( a _{0} ) + cos ( a _{0} ) ( cos ( a _{0} ) + 1 ) ) )}{( 2 + 2 cos ( a _{0} ) ) ^{2}}

y = \frac{cos ( a _{0} ) ( 2 + 2 cos ( a _{0} ) ) + 2 sin ^{2} ( a _{0} )}{( 2 + 2 cos ( a _{0} ) ) ^{2}} t + \frac{2 ( sin ( a _{0} ) ( cos ( a _{0} ) + 1 ) - a _{0} ( sin ^{2} ( a _{0} ) + cos ( a _{0} ) ( cos ( a _{0} ) + 1 ) ) )}{( 2 + 2 cos ( a _{0} ) ) ^{2}}

What is the tangent of (sin(t))/(2+2cos(t)) ?
The tangent of (sin(t))/(2+2cos(t)) is