Double-angle tangent and compound-angle cosine from two right-angled triangles on a common altitude
Double-angle tangent and compound-angle cosine from two right-angled triangles on a common altitude
IB Year 6 | Grade 12
7 marks
In the diagram, \(N\) is the point on the line segment \(QR\) such that \(PN\) is perpendicular to \(QR\). It is given that \(PN=2\), \(QN=1\) and \(NR=3\), and that \(\angle RPN=\alpha\) and \(\angle QPN=\beta\).
- Find the exact value of \(\tan2\alpha\).[2]
- Find the exact value of \(\cos(\alpha+\beta)\).[3]
- Find the exact value of \(\cos\left(2\arctan\frac12\right)\).[2]
Solution:
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Answer:(a) \(-\frac{12}{5}\) (b) \(\frac{\sqrt{65}}{65}\) (c) \(\frac35\)
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