Geometry

Fundamental Trigonometric Identities

Fundamental Trigonometric Identities - Problem Solving

         

If tanθ=27 \tan \theta = \frac{ 2}{7} , what is the value of

cosθ+3sinθcosθ3sinθ? \frac{ \cos \theta + 3 \sin \theta } { \cos \theta - 3 \sin \theta}?

Details and assumptions

You may read up on Trigonometric Functions.

Given that sinθ+cosθ=109\sin \theta + \cos \theta = \frac{10}{9} and sin(2θ)=ab\sin (2\theta) = \frac{a}{b}, where aa and bb are coprime positive integers, what is the value of a+ba + b?

If

i=090cos2i=ab, \sum_{i=0} ^ {90} \cos ^2 i ^ \circ = \frac {a}{b},

where aa and bb are positive coprime integers what is the value of a+ba+b?

PP is a point on a semicircle with diameter AB=10\overline{AB}=10. What is the maximum value of 3AP+4BP?3\overline{AP}+4\overline{BP}?

What is the value of:

(11cos72)(1+1sin18)(11sin72)(1+1cos18)? \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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