Question

If \(\tan \theta=\frac{3}{4}\) and \(\theta \in\left(180^{\circ} ; 360^{\circ}\right)\) calculate without the use of a calculator and with the aid of a diagram the value of:

(a) \(\sin 2 \theta\)
(b) \(\cos 2 \theta\)
(c) \(\tan 2 \theta\)

Answer 1

(a) \(\sin 2 \theta=2 \sin \theta \cdot \cos \theta\)
\(\therefore \sin 2 \theta=2 \cdot\left(\frac{-3}{5}\right)\left(\frac{-4}{5}\right)\)
\(\therefore \sin 2 \theta=\frac{24}{25}\)

 

(b) \(\cos 2 \theta=2 \cos ^2 \theta-1\)
\[
\begin{aligned}
& \therefore \cos 2 \theta=2\left(\frac{-4}{5}\right)^2-1 \\
& =\frac{32}{25}-1=\frac{7}{25}
\end{aligned}
\]

 

(c) \(\tan 2 \theta=\dfrac{\sin 2 \theta}{\cos 2 \theta}\)
\(\therefore \tan 2 \theta=\dfrac{\dfrac{24}{25}}{\dfrac{7}{25}}=\dfrac{24}{7}\)