Fundamental Trigonometric Identities
If \( \tan \theta = \frac{ 2}{7} \), what is the value of
\[ \frac{ \cos \theta + 3 \sin \theta } { \cos \theta - 3 \sin \theta}? \]
Details and assumptions
You may read up on Trigonometric Functions.
Given that \(\sin \theta + \cos \theta = \frac{10}{9}\) and \(\sin (2\theta) = \frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, what is the value of \(a + b\)?
If
\[ \sum_{i=0} ^ {90} \cos ^2 i ^ \circ = \frac {a}{b}, \]
where \(a\) and \(b\) are positive coprime integers what is the value of \(a+b\)?
\(P\) is a point on a semicircle with diameter \(\overline{AB}=10\). What is the maximum value of \[3\overline{AP}+4\overline{BP}?\]
What is the value of:
\( \left(1-\frac{1}{\cos 72 ^\circ}\right)\left(1+\frac{1}{\sin 18 ^\circ}\right)\left(1-\frac{1}{\sin 72 ^\circ}\right)\left(1+\frac{1}{\cos 18 ^\circ}\right)? \)