Question
(i) Express \(3\cos \Theta +\sin \Theta \) in the form \(R\cos (\Theta -\alpha ) \), where R > 0 and \(0^{\circ}< \alpha < 90^{\circ}\), giving the exact value of R and the value of correct to 2 decimal places.
(ii) Hence solve the equation
3 cos 2x + sin 2x = 2,
giving all solutions in the interval \(0^{\circ}\leq x\leq 360^{\circ}\).
Answer/Explanation
(i) State \(R=\sqrt{10}\)
Use trig formula to find α
Obtain α = 18.43 with no errors seen
(ii) Carry out evaluation of \(\cos ^{-1}\left ( \frac{2}{R} \right )\approx 50.77^{\circ}\)
Carry out correct method for one correct answer
Obtain one correct answer e.g. 34.6°
Carry out correct method for a further answer
Obtain remaining 3 answers 163.8°, 214.6°, 343.8° and no others in the range
Question
(i) Express 5 sin 2θ + 2 cos 2θ in the form Rsin(2θ + α), where R > 0 and 0◦ < α < 90◦, giving the exact value of R and the value of α correct to 2 decimal places.
Hence
(ii) solve the equation
5 sin 2θ + 2 cos 2θ = 4,
giving all solutions in the interval 0◦ ≤ θ ≤ 360◦
(iii) determine the least value of \(\frac{1}{(10\sin 2\Theta +4\cos 2\Theta )^{2}}\) as θ varies.
Answer/Explanation
(i) State \(R=\sqrt{29}\)
Use trig formula to find α
Obtain \(\alpha =21.80^{\circ}\) with no errors seen
(ii) Carry out evaluation of \(\sin ^{-1}(\frac{4}{R})\approx 49.97^{\circ}\)
Carry out correct method for one correct answer
Obtain one correct answer e.g. \(13.1^{\circ}\)
Carry out correct method for a further answer
Obtain remaining 3 answers \(55.1^{\circ}\),\( 193.1^{\circ}\),\(235.1^{\circ}\)and no others in the range
(iii) Greatest value of \(10\sin 2\Theta +4\cos 2\Theta =2\sqrt{29}\)
\(\frac{1}{116}\)
Question
(i)Show that \( \frac{\sin \Theta }{\sin \Theta +\cos \Theta }+\frac{\cos \Theta }{\sin \Theta -\cos \Theta }=\frac{1}{\sin ^{2}\Theta -\cos ^{2}\Theta }\).
(ii)Hence solve the equation \(\frac{\sin \Theta }{\sin \Theta +\cos \Theta }+\frac{\cos \Theta }{\sin \Theta -\cos \Theta }=3\) for \(0^{\circ}\leq \Theta \leq 360^{\circ}\)
Answer/Explanation
(i)\(\frac{\sin \Theta\left ( \sin \Theta -\cos \Theta \right )+\cos \Theta \left ( \sin \Theta +\cos \Theta \right ) }{\left ( \sin \Theta +\cos \Theta \right )\left ( \sin \Theta -\cos \Theta \right )}\)
\(\frac{\sin^{2}\Theta -\sin \Theta \cos \Theta +\sin \Theta \cos \Theta +\cos ^{2}\Theta }{\sin ^{2}\Theta -\cos ^{2}\Theta }\)
\(\frac{1}{\sin ^{2}\Theta -\cos ^{2}\Theta }\)
(ii)\(\sin ^{2}\Theta -\left ( 1-\sin ^{2} \Theta \right )=\frac{1}{3}\) or \(1-\cos ^{2}\Theta -\cos ^{2}\Theta =\frac{1}{3}\)
or \(3\left ( \sin ^{2} \Theta -\cos ^{2}\Theta \right )=\cos ^{2}\Theta +\sin ^{2}\Theta \)
\(\sin \Theta =\pm \sqrt{\frac{2}{3}}\) or \(\cos \Theta =\pm \sqrt{\frac{1}{3}}\)
\(\tan \Theta =\pm \sqrt{2}\)
\(\Theta =54.7^{\circ},125.3^{\circ},234.7^{\circ},305.3^{\circ}\)