3.7.1 Reciprocal Trig Functions

What are the reciprocal trig functions?

• There are three reciprocal trig functions that each correspond to either sin, cos or tan
  • Secant (sec x)
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  • Cosecant (cosec x)
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  • Cotangent (cot x)
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  • The identities above for sec x and cosec x are given in the formula booklet
  • The identity for cot x is not given, you will need to remember it
  • A good way to remember which function is which is to look at the third letter in each of the reciprocal trig functions
    • cot x is 1 over tan x etc
• Each of the reciprocal trig functions are undefined for certain values of x
  • sec x is undefined for values of x for which cos x = 0
  • cosec x is undefined for values of x for which sin x = 0
  • cot x is undefined for values of x for which tan x = 0
    • When tan x is undefined, cot x = 0
• Rearranging the identity gives
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    • This is not in the formula booklet but is easily derived
• Be careful not to confuse the reciprocal trig functions with the inverse trig functions
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What do the graphs of the reciprocal trig functions look like?

• The graph of y = secx has the following properties:
• The y-axis is a line of symmetry
• It has a period of 360° (2π radians)
• There are vertical asymptotes wherever cos x = 0
• If drawing the graph without the help of a GDC it is a good idea to sketch cos x first and draw these in
• The domain is all x except odd multiples of 90° (90°, -90°, 270°, -270°, etc.)
• in radians this is all x except odd multiples of π/2 (π/2, - π/2, 3π/2, -3π/2, etc.)
• The range is y ≤ -1 or y ≥ 1

• The graph of y = cosec x has the following properties:
• It has a period of 360° (2π radians)
• There are vertical asymptotes wherever sin x = 0
• If drawing the graph it is a good idea to sketch sin x first and draw these in
• The domain is all x except multiples of 180° (0°, 180°, -180°, 360°, -360°, etc.)
• in radians this is all x except multiples of π (0, π, - π, 2π, -2π, etc.)
• The range is y ≤ -1 or y ≥ 1

•  The graph of y = cot x has the following properties
• It has a period of 180° or π radians
• There are vertical asymptotes wherever tan x = 0
• The domain is all x except multiples of 180° (0°, 180°, -180°, 360°, -360°, etc.)
• In radians this is all x except multiples of π (0, π, - π, 2π, -2π, etc.)
• The range is y ∈ ℝ (i.e. cot can take any real number value)

What are the Pythagorean Identities?

• Aside from the Pythagorean identity sin²x + cos²x = 1 there are two further Pythagorean identities you will need to learn
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• Both can be found in the formula booklet
• Both of these identities can be derived from sin²x + cos²x = 1 
• To derive the identity for sec²x divide sin²x + cos²x = 1 by cos²x
• To derive the identity for cosec²x divide sin²x + cos²x = 1 by sin²x