Help? Maths simplification problem?

Just a little stuck with a particular C4 simplification problem. If anyone could help me out I would appreciate it :smile:

:

X = 10 cosϴ + 5 cos2ϴ,
Y = 10 sinϴ + 5 sin2ϴ.

Show that X² + Y² = 125 + 100 cosϴ.

No matter how many times I try, I can't seem to come that answer. Ive got as far as:

X² + Y² = 100 cos²ϴ + 100 cosϴcos2ϴ + 25 cos²2ϴ + 100 sin²ϴ + 100 sinϴsin2ϴ + 25 sin²2ϴ.

After that im clueless as to how to get to 125 + 100 cosϴ even when using the trig identities to replace cos2ϴ and sin2ϴ.

Reply 1

Radioactivedecay

16

Try using the fact that sin^2 + cos^2 =1, and simplify from there.

Reply 2

brainmaster

14

Original post by aaabattery

Just a little stuck with a particular C4 simplification problem. If anyone could help me out I would appreciate it :smile:

:

X = 10 cosϴ + 5 cos2ϴ,
Y = 10 sinϴ + 5 sin2ϴ.

Show that X² + Y² = 125 + 100 cosϴ.

No matter how many times I try, I can't seem to come that answer. Ive got as far as:

X² + Y² = 100 cos²ϴ + 100 cosϴcos2ϴ + 25 cos²2ϴ + 100 sin²ϴ + 100 sinϴsin2ϴ + 25 sin²2ϴ.

After that im clueless as to how to get to 125 + 100 cosϴ even when using the trig identities to replace cos2ϴ and sin2ϴ.

what you want is in terms of cos so obviously you have to turn all sin values to cos....meaning you will have to turn Y^2 in terms of cos which will be;
Y^2 = 100 (1-cos^2 x) + 100 sin x (2sin x cos x) +25 (1-cos^2x)

now expand this and you put this Y^2 in the equation X^2 + Y^2 and you will realise that two terms will cancel each other.

later you will have one with cos 2x and sin^2 x...again write this in terms of cos I.e cos 2x = 2cos....... and simplify and you will see you will end up with 125-100cos x.

let me known if you are able to do it or now :smile:

Reply 3

Katarinanokat

12

The previous poster is right. You should move things as such:

100cos²x + 100sin²x + 25cos²x + 25sin²x + 100cosxcos2x + 100sinxsin2x

You then get:

100 + 25 + 100( cosxcos2x + sinxsin2x)

We can use compound identities of cos, such that Cos (A -B) = cosAcosB + sinAsinB

so:
125 + 100 cos( 2x-x)
125+ 100cosx

Reply 4

aaabattery

OP

11

I dont know what im doing wrong:

X² + Y² = 100 cos²ϴ + 100 cosϴcos2ϴ + 25 cos²2ϴ + 100 sin²ϴ + 100 sinϴsin2ϴ + 25 sin²2ϴ
X² + Y² = 100 + 100 cosϴcos2ϴ + 25 cos²2ϴ + 100 sinϴsin2ϴ + 25 sin²2ϴ
X² + Y² = 100 + 100 cosϴ(2 cos²ϴ - 1) + 25 (2 cos²ϴ - 1)² + 100 sinϴ(2 sinϴcosϴ) + 25 (2 sinϴcosϴ)²
X² + Y² = 100 + 100 cosϴ(2 cos²ϴ - 1) + 25 (4 cos⁴ϴ - 4 cos²ϴ + 1) + 100 sinϴ(2 sinϴcosϴ) + 25 (4sin²ϴcos²ϴ)
X² + Y² = 100 + 200 cos³ϴ - 100 cosϴ + 100 cos⁴ϴ - 100 cos²ϴ + 25 + 200 sin²ϴcosϴ + 100 sin²ϴcos²ϴ

ummm.. what now?

Reply 5

aaabattery

OP

11

Original post by Katarinanokat

The previous poster is right. You should move things as such:

100cos²x + 100sin²x + 25cos²x + 25sin²x + 100cosxcos2x + 100sinxsin2x

You then get:

100 + 25 + 100( cosxcos2x + sinxsin2x)

We can use compound identities of cos, such that Cos (A -B) = cosAcosB + sinAsinB

so:
125 + 100 cos( 2x-x)
125+ 100cosx

Actually you're method makes perfect sense rather than trying to substitute cos2ϴ and sin2ϴ. Thank you so much thats perfect! And thank you everyone else too for helping me out with my maths :smile:

(edited 6 years ago)

Help? Maths simplification problem?

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