Standard +0.8 This question requires students to manipulate the compound angle formula for tan(θ-φ), set up a system of equations involving tan θ and tan φ, and solve simultaneously while considering the domain restrictions. It goes beyond routine application of formulae, requiring algebraic manipulation and careful consideration of multiple solutions within the given range.
3 The angles \(\theta\) and \(\phi\) lie between \(0 ^ { \circ }\) and \(180 ^ { \circ }\), and are such that
$$\tan ( \theta - \phi ) = 3 \quad \text { and } \quad \tan \theta + \tan \phi = 1$$
Find the possible values of \(\theta\) and \(\phi\).
Use \(\tan(A \pm B)\) and obtain an equation in \(\tan\theta\) and \(\tan\phi\)
M1*
Substitute throughout for \(\tan\theta\) or for \(\tan\phi\)
DM1
Obtain \(3\tan^2\theta - \tan\theta - 4 = 0\) or \(3\tan^2\phi - 5\tan\phi - 2 = 0\), or 3-term equivalent
A1
Treat answers in radians as a misread. Ignore answers outside the given interval. [SR: Two correct values of \(\theta\) (or \(\phi\)) score A1; then A1 for both correct \(\theta\), \(\phi\) pairs.]
3 The angles $\theta$ and $\phi$ lie between $0 ^ { \circ }$ and $180 ^ { \circ }$, and are such that
$$\tan ( \theta - \phi ) = 3 \quad \text { and } \quad \tan \theta + \tan \phi = 1$$
Find the possible values of $\theta$ and $\phi$.\\
\hfill \mbox{\textit{CAIE P3 Q3 [6]}}