If cot \(\theta\) = \(\frac{x}{y}\), find cosec\(\theta\)
A. \(\frac{1}{y}\)(x
2 + y
2)
B. \(\frac{x}{y}\)
C. \(\frac{1}{y}\)\(\sqrt{x^2 + y^2}\)
D. \(\frac{x - y}{y}\)
Correct Answer: C
Explanation
\(\cot \theta = \frac{x}{y}\)
\(\implies \tan \theta = \frac{y}{x}\)
\(opp = y; adj = x\)
Using Pythagoras theorem, \(Hyp^{2} = Opp^{2} + Adj^{2}\)
\(Hyp^{2} = y^{2} + x^{2}\)
\(Hyp = \sqrt{y^{2} + x^{2}}\)
\(\sin \theta = \frac{y}{\sqrt{y^{2} + x^{2}}}\)
\(\therefore \csc \theta = \frac{\sqrt{y^{2} + x^{2}}}{y}\)
= \(\frac{1}{y}(\sqrt{y^{2} + x^{2}})\)
