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Geometry

Fundamental Trigonometric Identities

Fundamental Trigonometric Identities - Problem Solving

         

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:

\( \left(1-\frac{1}{\cos 72 ^\circ}\right)\left(1+\frac{1}{\sin 18 ^\circ}\right)\left(1-\frac{1}{\sin 72 ^\circ}\right)\left(1+\frac{1}{\cos 18 ^\circ}\right)? \)

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