Trigonometry

Synopsis

Important Concepts

• An angle which has a magnitude as well as the direction of rotation is known as directed angle.
• An angle is formed by two rays called initial ray and terminal ray, with a common initial point called vertex.
• If the rotation from the initial ray to the terminal ray is anti-clockwise it is to be taken as positive and if it is clockwise, rotation is to be taken as negative.
• Each trigonometrical ratio is a real number and has no units.

Trigonometric Functions (or ratios)

• In a right-angle triangle ABC, let  ABC = Ѳ
  
  Let AB = x, BC = y and AC = r.
  Then we define the trigonometric ratios as under
  
  

Sign of trigonometric functions in various quadrants

• In quadrant I, all the trigonometric functions are positive. 
• In quadrant II, only sine and cosec functions are positive. 
• In quadrant III, only tan and cot functions are positive. 
• In quadrant IV, only cosine and sec functions are positive. 
• This is depicted as follows:
  
  

• In quadrants, where the y-axis is positive (i.e. I and II), sine is positive, and in quadrants where the x-axis is positive (i.e. I and IV), cosine is positive.
• A simple rule to remember the sign of the trigonometrical ratios, in all the four quadrants, is the four letter phrase—All School To College.

• Relations between Trigonometric Ratios
• Reciprocal relation: 
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• Quotient relation: 
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