Cambridge Additional Mathematics

Example 16 Self Tutor

¼ 0

O

7 ¼

6

11 ¼

6

-_Qw

O

A N

P

1

1

M

q 2 q

q

O B

Solve for 06 x 62 ¼:

a 2 sin^2 x+ sinx=0 b 2 cos^2 x+ cosx¡1=0

a 2 sin^2 x+ sinx=0

) sinx(2 sinx+1)=0

) sinx=0or¡^12

sinx=0when

x=0,¼,or 2 ¼

sinx=¡^12 when

x=^76 ¼or^116 ¼

The solutions are: x=0,¼,^76 ¼,^116 ¼,or 2 ¼.

250 Trigonometric functions (Chapter 9)

2 Write down any discoveries from your table of values in 1.

3 In the diagram alongside, the semi-circle has radius 1 unit,

and PABb =μ.

AbPO=μ f4AOP is isoscelesg

PONb =2μ fexterior angle of a triangleg

a Find in terms ofμ, the lengths of:

i OM ii AM iii ON iv PN

b Use 4 ANP and the lengths inato show that:

i cosμ=

sin 2μ

2 sinμ

ii cosμ=

1 + cos 2μ

2 cosμ

c Hence deduce that:

i sin 2μ= 2 sinμcosμ ii cos 2μ= 2 cos^2 μ¡ 1

4 Starting with cos 2μ= 2 cos^2 μ¡ 1 , show that:

a cos^2 μ=^12 +^12 cos 2μ b sin^2 μ=^12 ¡^12 cos 2μ

Sometimes we may be given trigonometric equations in quadratic form.

For example, 2 sin^2 x+ sinx=0 and 2 cos^2 x+ cosx¡1=0 are quadratic equations where the

variables are sinx and cosx respectively.

These equations can be factorised by quadratic factorisation and then solved forx.

G Trigonometric equations in quadratic form

FORM

G

The double angle formulae are

not required for the syllabus

but are very useful.

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Additional Mathematics
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