Trigonometric Functions

Trigonometrical Functions

  • Angle : Angle is a measure of rotation of a given ray about its initial point. The original ray is called the initial side and the final ray after rotation is called terminal side of the angle. The point of rotation is called the vertex.
  • Relation between degree and radianSince a circle subtends at the centre an angle whose radian is 2π and its degree measure is 360º , it follows that   2π radian = 360º   or  π radian = 180º.
  • Trigonometric Functions : The extension of trigonometric ratios to any angle in terms of radian measure and study them as trigonometric functions.

Part1: View the Topic in this video From 39:36 To 56:00

Part2: View the Topic in this video From 00:45 To 38:27

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1. Relation between trigonometric ratios (functions)

      a) sin θ . cosec θ = 1
      b) tan θ . cot θ = 1
      c) cos θ . sec θ = 1
      d) \tt \tan \theta = \frac{\sin \theta}{\cos \theta}
      e) \tt \cot \theta = \frac{\cos \theta}{\sin \theta}

2. Fundamental trigonometric identities

      a) sin2 θ + cos2 θ = 1
      b) 1 + tan2 θ = sec2 θ
      c) 1 + cot2 θ = cosec2 θ

3. Trigonometric Ratios of Some Standard Angles

Angle  0º  30º  45º  60º 90º  120º  135º  150º  180º 
sin  \tt \frac {1}{2}  \tt \frac {1}{\sqrt{2}}  \tt \frac {\sqrt{3}}{2}  \tt \frac {\sqrt{3}}{2}  \tt \frac {1}{\sqrt{2}}  \tt \frac {1}{2} 
cos  \tt \frac {\sqrt{3}}{2}  \tt \frac {1}{\sqrt{2}}  \tt \frac {1}{2}  \tt -\frac {1}{2}  \tt -\frac {1}{\sqrt{2}}  \tt -\frac {\sqrt{3}}{2}  −1
tan  \tt \frac {1}{\sqrt{3}}  \tt \sqrt{3}  ∞  \tt -\sqrt{3}  −1  \tt -\frac {1}{\sqrt{3}} 
cot  ∞  \tt \sqrt{3}  \tt \frac{1}{\sqrt{3}}  \tt -\frac{1}{\sqrt{3}}  −1  \tt -\sqrt{3}  −∞ 
sec  \tt \frac{2}{\sqrt{3}}  \tt \sqrt{2}  ∞  −2  \tt -\sqrt{2}  \tt -\frac{2}{\sqrt{3}}  −1 
cosec  ∞  \tt \sqrt{2}  \tt \frac{2}{\sqrt{3}}  \tt \frac{2}{\sqrt{3}}  \tt \sqrt{2}  ∞ 

 4. Trigonometric Ratios of Some Special Angles

Angle \tt 7\frac{1^{o}}{2}  15º  \tt 22\frac{1^{o}}{2}  18º  36º 
sin θ  \tt \frac{\sqrt{4-\sqrt{2}-\sqrt{6}}}{2\sqrt{2}}  \tt \frac{\sqrt{3}-1}{2\sqrt{2}}  \tt \frac{1}{2}\sqrt{2-\sqrt{2}}  \tt \frac{\sqrt{5}-1}{4}  \tt \frac{1}{4}\sqrt{10-2\sqrt{5}} 
cos θ  \tt \frac{\sqrt{4+\sqrt{2}+\sqrt{6}}}{2\sqrt{2}}  \tt \frac{\sqrt{3}+1}{2\sqrt{2}}  \tt \frac{1}{2}\sqrt{2+\sqrt{2}}  \tt \frac{1}{4}\sqrt{10+2\sqrt{5}}  \tt \frac{\sqrt{5}+1}{4} 
tan θ  \tt \left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{2}-1\right)  \tt 2-\sqrt{3}  \tt \sqrt{2}-1  \tt \frac{\sqrt{25-10\sqrt{15}}}{5} \tt \sqrt{5-2\sqrt{5}}