Mathematics

Problem

An acute angle

\theta

satisfies

\tan\theta = \dfrac{2k + 1}{3}

, where

k

is a real parameter. Answer the following questions. a. (i) Derive expressions for

\sin\theta

and

\cos\theta

in terms of

k

, leaving each in simplest radical form. b. (ii) If

\cos\theta = \dfrac{5}{\sqrt{34}}

, determine the value of

k

. c. (iii) Using the value of

k

found in part (ii), state

\sin\theta

and verify that

\sin^{2}\theta + \cos^{2}\theta = 1