3.2 Trigonometric Functions: The Unit Circle Approach

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Solved Examples

  1. Locate xx: III Quadrant   ⟹  tan⁡,cot⁡\implies \tan, \cot positive; others negative.
  2. Find sin⁡x\sin x: sin⁡2x+cos⁡2x=1  ⟹  sin⁡2x=1−925=1625\sin^2 x + \cos^2 x = 1 \implies \sin^2 x = 1 - \frac{9}{25} = \frac{16}{25} Since III quad, sin⁡x=−45\sin x = -\frac{4}{5}.
  3. Others:
    • tan⁡x=sin⁡xcos⁡x=−4/5−3/5=43\tan x = \frac{\sin x}{\cos x} = \frac{-4/5}{-3/5} = \frac{4}{3}
    • cot⁡x=34\cot x = \frac{3}{4}
    • sec⁡x=−53\sec x = -\frac{5}{3}
    • cosec x=−54\text{cosec } x = -\frac{5}{4}

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