Two angles with tan relationships

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 \(\beta\).

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 \(\phi\).

6 The angles \(A\) and \(B\) are such that $$\sin \left( A + 45 ^ { \circ } \right) = ( 2 \sqrt { } 2 ) \cos A \quad \text { and } \quad 4 \sec ^ { 2 } B + 5 = 12 \tan B$$ Without using a calculator, find the exact value of \(\tan ( A - B )\).

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 \(\phi\).

8

1. Given that \(\tan A = t\) and \(\tan ( A + B ) = 4\), find \(\tan B\) in terms of \(t\).
2. Solve the equation $$2 \tan \left( 45 ^ { \circ } - x \right) = 3 \tan x$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

5 The angles \(\alpha\) and \(\beta\) lie between \(0 ^ { \circ }\) and \(180 ^ { \circ }\) and are such that $$\tan ( \alpha + \beta ) = 2 \quad \text { and } \quad \tan \alpha = 3 \tan \beta .$$ Find the possible values of \(\alpha\) and \(\beta\).

3．The angle \(\theta , 0 < \theta < \frac { \pi } { 2 }\) ，satisfies $$\tan \theta \tan 2 \theta = \sum _ { r = 0 } ^ { \infty } 2 \cos ^ { r } 2 \theta$$ （a）Show that \(\tan \theta = 3 ^ { p }\) ，where \(p\) is a rational number to be found．
（b）Hence show that \(\frac { \pi } { 6 } < \theta < \frac { \pi } { 4 }\)