Standard +0.8 This question requires students to manipulate the tan addition formula algebraically, set up a system involving tan(θ-φ) and individual tan values, then solve for two unknowns with domain restrictions. It goes beyond routine formula application, requiring strategic algebraic manipulation and consideration of multiple cases within the given range, making it moderately challenging but not exceptional.
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 \(\phi\).
[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.]
Use $\tan(A \pm B)$ and obtain an equation in $\tan \theta$ and $\tan \phi$ | M1* |
Substitute throughout for $\tan \theta$ or for $\tan \phi$ | dep M1* |
Obtain $3\tan^2 \theta - \tan \theta - 4 = 0$ or $3\tan^2 \phi - 5\tan \phi - 2 = 0$, or 3-term equivalent | A1 |
Solve a 3-term quadratic and find an angle | M1 |
Obtain answer $\theta = 135°, \phi = 63.4°$ | A1 |
Obtain answer $\theta = 53.1°, \phi = 161.6°$ | A1 |
| [6] |
[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 2015 Q3 [6]}}