Aims:• To know the relationship between the graphs and notation of

cosine, sine and tan, with secant, cosecant and cotangent .• To be able to state the domain and range of the new

trigonometric functions

• To be able to calculate exact values of these reciprocal trig functions, with use of a calculator.

Trigonometry Lesson 2

The reciprocal trigonometric functions The reciprocal trigonometric functions are cosecant, secant and cotangent.

They are related to the three main trigonometric ratios as follows:

This is short for cosecant.

1sin x

This is short for secant.

1cos x

This is short for cotangent.

1tan x

Notice that the first letter of sin, cos and tan happens to be the same as the third letter of the corresponding reciprocal functions cosec, sec and cot.

The graph of sec x

State the range and domain of sec x

What is the wave length (period) of cos x? __________

The graph of cosec x

State the range and domain of cosec x

What is the wave length (period) of sin x? __________

The graph of cot x

State the range and domain of cot x

What is the wave length (period) of tan x? __________

Problems involving reciprocal trig functions Use a calculator to find, to 2 d.p., the value of:a) sec 85° b) cosec 220° c) cot –70°

a) sec 85° =1

cos85

b) cosec 220° =1

sin220

c) cot –70° =1

tan 70

Find the exact value of:a) cosec b) cot c) sec –6

23 3

4

a)

b)

c)

You can put your calculator in radians mode and recognise those special decimals!

6sin

1

32tan

1

43cos

1

Problems involving reciprocal trig functions Example 1Simplify

x2

cosec

Do questions 1-4 from ex B page 57

Example 2Simplify sec x cot x