The acute angle x radians is such that \(\tan x = k\), where \(k\) is a positive constant. Express, in terms of \(k\),
- \(\tan(\pi - x)\),
- \(\tan\left(\frac{1}{2}\pi - x\right)\),
- \(\sin x\).
Solution
(i) Using the identity \(\tan(\pi - x) = -\tan x\), we have:
\(\tan(\pi - x) = -k\).
(ii) Using the identity \(\tan\left(\frac{\pi}{2} - x\right) = \cot x = \frac{1}{\tan x}\), we have:
\(\tan\left(\frac{\pi}{2} - x\right) = \frac{1}{k}\).
(iii) Using the identity \(\sin x = \frac{\tan x}{\sqrt{1 + \tan^2 x}}\), we have:
\(\sin x = \frac{k}{\sqrt{1 + k^2}}\).
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