Standard +0.8 This question requires applying the tan addition formula to create an equation in tan β, then solving a quadratic. Students must also handle the constraint that angles lie in (0°, 180°) and determine which solutions are valid. The multi-step algebraic manipulation and angle range considerations elevate this above routine formula application, but it remains a standard P3-level problem without requiring exceptional insight.
4 The angles \(\alpha\) and \(\beta\) lie in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\), and are such that
$$\tan \alpha = 2 \tan \beta \quad \text { and } \quad \tan ( \alpha + \beta ) = 3 .$$
Find the possible values of \(\alpha\) and \(\beta\).
Use \(\tan(A \pm B)\) formula and obtain an equation in \(\tan \alpha\) and \(\tan \beta\)
M1*
Substitute throughout for \(\tan \alpha\) or for \(\tan \beta\)
M1(dep*)
Obtain \(2\tan^2\beta + \tan\beta - 1 = 0\) or tan \(\alpha + \tan\alpha - 2 = 0\), or equivalent
A1
Solve a 3-term quadratic and find an angle
M1
Obtain answer \(\alpha = 45°, \beta = 26.6°\)
A1
Obtain answer \(\alpha = 116.6°, \beta = 135°\)
A1
[6]
[Treat answers given in radians as misread. Ignore answers outside the given range.]
[SR: Two correct values of \(\alpha\) (or \(\beta\)) score A1; then A1 for both correct \(\alpha, \beta\) pairs]
Use $\tan(A \pm B)$ formula and obtain an equation in $\tan \alpha$ and $\tan \beta$ | M1* |
Substitute throughout for $\tan \alpha$ or for $\tan \beta$ | M1(dep*) |
Obtain $2\tan^2\beta + \tan\beta - 1 = 0$ or tan $\alpha + \tan\alpha - 2 = 0$, or equivalent | A1 |
Solve a 3-term quadratic and find an angle | M1 |
Obtain answer $\alpha = 45°, \beta = 26.6°$ | A1 |
Obtain answer $\alpha = 116.6°, \beta = 135°$ | A1 | [6]
[Treat answers given in radians as misread. Ignore answers outside the given range.]
[SR: Two correct values of $\alpha$ (or $\beta$) score A1; then A1 for both correct $\alpha, \beta$ pairs]
4 The angles $\alpha$ and $\beta$ lie in the interval $0 ^ { \circ } < x < 180 ^ { \circ }$, and are such that
$$\tan \alpha = 2 \tan \beta \quad \text { and } \quad \tan ( \alpha + \beta ) = 3 .$$
Find the possible values of $\alpha$ and $\beta$.
\hfill \mbox{\textit{CAIE P3 2009 Q4 [6]}}