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Fundamental Trigonometric Identities

These are your basic building blocks for solving trigonometric equations and understanding how the pieces fit together. Using these identities can make sense of even the scariest looking trig.

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 \(a + b\)?


\[ \sum_{i=0} ^ {90} \cos ^2 i ^ \circ = \frac {a}{b}, \]

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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