Prelim 2016 Maths Help Thread (1 Viewer)

[InteGrand]

If sin (theta)=3/4, 90 degrees is smaller than (theta) is smaller than 180 degrees, evaluate in surd form,

a. Sin2(theta)
b. cos2(theta)
c.tan2(theta)
In which quadrant does 2(theta) lie in?

The others are quite similar in nature.

 

[eyeseeyou]

The others are quite similar in nature.

Thnx

 

[eyeseeyou]

I do not know what I am doing wrong for the following questions:

a. sin4xcos4x
b. 1+cos(180+2(theta))
c.sinxcosxcos2x
d. 2sin2xcos2x

help me please

 

[eyeseeyou]

(2tantheta)/(1-tan squared theta) when theta = 22.5 degrees

sin squared 50+sin squared 40

and help me with this one as well

 

[InteGrand]

(2tantheta)/(1-tan squared theta) when theta = 22.5 degrees

sin squared 50+sin squared 40

and help me with this one as well

Both are clearly equal to 1 (can you see why?).

 

[eyeseeyou]

Both are clearly equal to 1 (can you see why?).

No not quite

 

[InteGrand]

No not quite

First one: double-angle formula for tan, noting tan(45 deg) = 1.

Second one: Complementary and Pythagorean trig. identities.

 

[eyeseeyou]

First one: double-angle formula for tan, noting tan(45 deg) = 1.

Second one: Complementary and Pythagorean trig. identities.

Thanks integrand

 

[eyeseeyou]

1. sin(45-x)cos(45-x)
2. (1-cos 2theta)/ (1+cos2theta)
3. 2cos squared 3x -1
4. (1-t^2)/(1+t^2) where t=tan theta/2

 

[InteGrand]

1. sin(45-x)cos(45-x)
2. (1-cos 2theta)/ (1+cos2theta)
3. 2cos squared 3x -1
4. (1-t^2)/(1+t^2) where t=tan theta/2

1) Use the identity sin(t)cos(t) = (1/2)*sin(2t).

2) Essentially proved here: http://community.boredofstudies.org...do-we-need-memorise-identity.html#post7159341

3) Double angle formula for cos

4) cos(theta)

 

[eyeseeyou]

1) Use the identity sin(t)cos(t) = (1/2)*sin(2t).

2) Essentially proved here: http://community.boredofstudies.org...do-we-need-memorise-identity.html#post7159341

3) Double angle formula for cos

4) cos(theta)

for 2 and 4 could you please be more clearer

 

[eyeseeyou]

1. Using double angle formula for tan (theta), find tan 22.5 in simplest surd form
2. If tan (alpha)=k tan (beta), show that
(k-1)sin(aplha +beta) = (k+1) sin (alpha - beta)

 

[InteGrand]

1. Using double angle formula for tan (theta), find tan 22.5 in simplest surd form
2. If tan (alpha)=k tan (beta), show that
(k-1)sin(aplha +beta) = (k+1) sin (alpha - beta)

1) Let t = tan(22.5 deg), then 2t/(1 – t^2) = tan(45 deg) = 1, by the double angle formula.

So 2t = 1-t^2 ==> t^2 +2t = 1.

So (t+1)^2 = 2

i.e. t = 1 + √(2) (taking the positive solution; the negative solution is 1 – √(2), which we know isn't the right one, since t > 0 as 22.5 deg is acute).

 

[leehuan]

Oops I got beaten

Tip: If you have cos(2x) by itself then it might be hard to figure out which to use.

But if you have 1-cos(2x) try to get used to rewriting as 2sin^2(x)
And 1+cos(2x) as 2cos^2(x)

 

[InteGrand]

for 2 and 4 could you please be more clearer

(The LaTeX may take a while to load, it seems to have been playing up on this site today.)

 

[eyeseeyou]

Tip: If you have cos(2x) by itself then it might be hard to figure out which to use.

But if you have 1-cos(2x) try to get used to rewriting as 2sin^2(x)
And 1+cos(2x) as 2cos^2(x)

I was stuck at first and then thought about double angle then when I tried double angle I got some weird answer

Anyways thanks leehuan

 

[leehuan]

I was stuck at first and then thought about double angle then when I tried double angle I got some weird answer

Anyways thanks leehuan

Just don't forget with cos you have three choices for the double angles. Reason is because of the Pythagorean identity giving you three versions

 

[eyeseeyou]

Just don't forget with cos you have three choices for the double angles. Reason is because of the Pythagorean identity giving you three versions

yeah I know but which is the best to take?

 

[eyeseeyou]

(sin2theta+1)/(cos2theta)=(costheta-sintheta)/(costheta+sintheta)=2sec2theta

 

[leehuan]

yeah I know but which is the best to take?

The rule of thumb is when you have 1-cos(2x) you end up with 2sin^2(x)
And 1+cos(2x) gives 2cos^2(x)

You'll nee those identities when you learn to integrate sin^2(x) as well in the future.

Otherwise when you have cos(2x) by itself there is no rule of thumb. There's more intuition required for those. e.g. difference of two squares may mean I leave it in the original form cos^2(x)-sin^2(x)