Mathematics

Problem

An acute angle

\theta

satisfies

0^{\circ}<\theta<90^{\circ}

and both

\sin\theta

and

\cos\theta

are non–zero. a. Express

\tan\theta

in terms of

\sin\theta

and

\cos\theta

. Explain why this expression is valid for all such angles. b. Suppose

\sin\theta=3k

and

\cos\theta=4k

for some positive real constant

k

. Show that the value of

\tan\theta

does not depend on

k

and find its value. c. Using the result of part (ii), interpret the numerical value of

\tan\theta

in the context of the sides of a right triangle having

\theta

as one of its acute angles.