Cambridge Additional Mathematics

The unit circle and radian measure (Chapter 8) 221

We define the reciprocal trigonometric functions cosecμ, secantμ, and cotangentμ as:

cosecμ=

1

sinμ

, secμ=

1

cosμ

, and cotμ=

1

tanμ

=

cosμ

sinμ

Using these definitions we can derive the identities:

tan^2 μ+ 1 = sec^2 μ and 1 + cot^2 μ= cosec^2 μ

Proof: Using sin^2 μ+ cos^2 μ=1,

sin^2 μ

cos^2 μ

+

cos^2 μ

cos^2 μ

=

1

cos^2 μ

fdividing each term bycos^2 μg

) tan^2 μ+ 1 = sec^2 μ

Also using sin^2 μ+ cos^2 μ=1,

sin^2 μ

sin^2 μ

+

cos^2 μ

sin^2 μ

=

1

sin^2 μ

fdividing each term bysin^2 μg

) 1 + cot^2 μ= cosec^2 μ

EXERCISE 8F

1 Without using a calculator, find:

a cosec

¡¼

3

¢

b cot

¡ 2 ¼

3

¢

c sec

¡ 5 ¼

6

¢

d cot (¼)

e cosec

¡ 4 ¼

3

¢

f sec

¡ 7 ¼

4

¢

2 Without using a calculator, find cosecx, secx, and cotx for:

a sinx=^35 , 06 x 6 ¼ 2 b cosx=^23 ,^32 ¼<x< 2 ¼

3 Find the otherfivetrigonometric ratios if:

a cosμ=^34 and^32 ¼<μ< 2 ¼ b sinx=¡^23 and ¼<x<^32 ¼

c secx=2^12 and 0 <x<¼ 2 d cosecμ=2 and ¼ 2 <μ<¼

e tan ̄=^12 and ¼< ̄<^32 ¼ f cotμ=^43 and ¼<μ<^32 ¼

4 Findallvalues ofμfor which:

a cosecμ is undefined b secμ is undefined

c cotμ is zero d cotμ is undefined.

Review set 8A

1 Convert these to radians in terms of¼:

a 120 ± b 225 ± c 150 ± d 540 ±

2 Find the acute angles that would have the same:

a sine as^23 ¼ b sine as 165 ± c cosine as 276 ±.

F Reciprocal trigonometric ratios

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_08\221CamAdd_08.cdr Monday, 23 December 2013 1:59:14 PM BRIAN