5.2a Six Trigonometric Functions of Any Angle (The definition of sin, cos, tan, cosec, sec and cot)

5.2a Six Trigonometric Functions of Any Angle 

(A) The definition of sin, cos, tan, cosec, sec and cot


1. Let P (x, y) be any point on the circumference of the circle with 
    centre O and of radius r. Based on ∆ OPQ in the above diagram,



2. The definitions of tangent, cotangent, secant and cosecant of any 
    angle are:




3. The relations of the trigonometric ratio of an angle θ with its
    complementary angle (90oθ) are:


For examples:
(a) sin 75o = cos (90o – 75o) = cos 15o
(b) tan 50o = cot (90o – 50o) = cot 40o
(c) sec 25o = cosec (90o – 25o) = cosec 65o


4. The trigonometric ratios of any negative angle (–θ) are:

  • A negative angle is an angle measured in a clockwise direction from the positive x-axis.
  • For example, – 60o is equivalent to 300o (360o – 60o).

Example:
Express each of the following trigonometric functions in terms of the trigonometric ratios of acute angles. Hence, find each value using a calculator.
(a) cos (– 325o)
(b) tan (– 124o)
(c) sin (– 115o)

Solution:
(a)
cos (– 325o)
= cos 325o      ← {The formula cos (–θ) = cos θ is used}
= cos (360o – 325o)     ← {At fourth quadrant, cos is positive}
= cos 35o
= 0.8192

(b) 
tan (– 124o)
= – tan 124o      ← {The formula tan (–θ) = – tan θ is used}
= – [– tan (180o – 124o)]     ← {At second quadrant, tan is negative}
= tan 56o
= 1.483

(c)
sin (– 115o)
= – sin 115o      ← {The formula sin (–θ) = – sin θ is used}
= – sin (180o – 115o)     ← {At second quadrant, sin is positive}
= – sin 65o
= – 0.9063