GCSE Mathematics Extension Material
SSM 1
Trigonometry 1
What you should know
opposite
adjacent
, cos T
, tan T
hypotenuse
hypotenuse
Pythagoras’ theorem: opposite2 + adjacent2 = hypotenuse2
opposite
adjacent
The trigonometric ratios: sin T
New idea
If you divide sin ș by cos ș you get
sin T opp adj
y
cos T hyp hyp
...
tan T
sin T
so tan T
.
cos T
sin2 ș means (sin ș)2 so you can write
2
§ opp ·
sin T ¨
¸ .
© hyp ¹
Using this idea you can add together sin2 ș
and cos2 ș to get
2
2
2
2
sin T cos T
§ opp · § adj ·
¨
¸ ¨
¸
© hyp ¹ © hyp ¹
...
2
1
so sin 2 T cos 2 T
1.
Task: Trigonometric identities
x
x
In both of the ideas above, what should the missing bits of algebra, represented by …,
be? (There is more than one line of algebra missing!)
sin T
Find the sine and cosine of some angles and check that, in every case, tan T
cos T
and sin2 ș + cos2 ș = 1.
Remember: sin2 ș means (sin ș)2 and cos2 ș means (cos ș)2 .
Take it further
x You can use these identities to find values for cos ș and tan ș even if you only know
sin ș. You don’t even need to find out what the angle ș is.
Try finding cos ș and tan ș when sin ș = 3 .
5
x
x
x
x
Can you find some other right-angled triangles where sin ș, cos ș and tan ș are all
rational numbers (fractions that can be written as one integer over another)?
Do these identities work when ș < 0° or ș > 90°?
Find out what secant, cosecant and cotangent are.
Try dividing sin2 ș + cos2 ș = 1 by cos2 ș. What identity do you get?
Where this goes next
At A level trigonometry is studied in Core Mathematics.
© MEI 2009