If sina = 4/5 calculate tan a/2. a is inth first quadrant. Deduct a relation for tan a/2

Before calculating tan a/2, we'll infer the formula for
tan a/2.

We know that the function tangent is a ratio of
sine and cosine functions.

tan a/2 = sin (a/2)/cos
(a/2)

We'll write the formula for the sine of
the half-angle.

sin (a/2) = sqrt [(1-cos
a)/2]

We'll write the formula for the cosine of
the half-angle.

cos (a/2) = sqrt [(1+cos
a)/2]

We'll make the
ratio:

sin (a/2)/cos(a/2) = sqrt [(1-cos a)/2]/sqrt [(1+cos
a)/2]

We'll simplify and we'll
get:

tan (a/2) = sqrt [(1-cos a)]/sqrt [(1+cos
a)]

We'll multiply by sqrt [(1+cos a)] to eliminate the
square root from denominator:

tan (a/2) = sqrt [(1-cos
a)]*sqrt [(1+cos a)]/(1+cos a)

tan (a/2) = sqrt [1-(cos
a)^2]/(1+cos a)

But 1-(cos a)^2 = (sin
a)^2

Since a is in the 1st quadrant, sqrt (sin a)^2 = sin
a

Now, we'll calculate tan (a/2) = sin a/(1+cos
a)

cos a = sqrt[1 - (sin
a)^2]

cos a = sqrt
(1-16/25)

cos a = 3/5

tan(a/2)
= (4/5)/(1 + 3/5)

tan(a/2) =
4/8

tan(a/2) =
1/2