VCE Methods Question Thread!

[darklight]

Can someone explain why 1/(absolute value of x) is the same thing as the absolute value of (1/x)?

[FlorianK]

ok so lets say
f(x)=1/x
g(x)=1/|x|
h(x)=|1/x|

Lets look at it graphically:
For f(x) we now that it is point symetric.

for g(x) we now that its the same as f(x) for all x value greater than 1 and for lower than 1 it is exactly the same as on the right side mirrored onto the left side.

for h(x) we now that its the same as f(x) and hence g(x) for all x-values greater than 1 and for lower than 1 it is exactly the 'left' part of f(x) mirrored over the x-axis and due to the point-symetry it is the same as the 'left' part of g(x)

[jadams]

also abs (1/x) = abs (1) / abs (x)

what is abs (1) ? 1.

therefore abs (1/x) equals 1/abs (x)

[PsychoM]

Why does cos(pi/2 - x ) = sin x?
Can someone explain why is this?

[brightsky]

draw the triangle corresponding to the angle x. now draw the triangle corresponding to the angle pi/2-x (if x is acute, this triangle should be in the first quadrant). we know that cos(angle) corresponds to the x-coordinate while sin(angle) corresponds to the y-coordinate. now the two triangles described above are congruent (AAS). this means that corresponding sides are equal in length. the rest should be self-explanatory...

hope this explanation's clear enough...hard to explain a diagram in words

[Planck's constant]

Why does cos(pi/2 - x ) = sin x?
Can someone explain why is this?

Picture a right angled triangle.
It has two angles (other than the 90 degree angle).
One of the angles is x, the other pi/2 -x
According to the definitions of sine and Cosine ratios (adjacent/hypotenuse etc), it should be obvious that the sine of one angle in the triangle equals the Cosine of the other and vice versa

[Phy124]

You can prove it to be true using the trigonometric addition formulae.

[Lasercookie]

I guess another way of looking at it is to sketch the functions cos(x) and sin(x) on the same set of axes.

http://i.imgur.com/XvVD0.png
http://i.imgur.com/RKePC.png

What's the difference between the two? They look pretty much the same, except they're shifted pi/2 units away from each other.

cos(pi/2-x) is just applying transformations to cos(x), shifting it by pi/2 units, to make it 'look' the same as the sin(x) function.

[brightsky]

the identity is basically true by definition. COsine is thus named because it is the COfunction of sine, which means cos(pi/2 - x) = sin(x). similarly, COtangent is the cofunction of tangent, which means cot(pi/2 - x) = tan(x).

[polar]

Consider a right-angled triangle:

[jimmy22]

Have a question troubling me.

P and Q are the points of intersection of the line y/2 + x/3 =1 with the x and y axes respectively. The gradient of QR is 1/2, where R is a point with x-coordinate 2a, a>0.

a) Find the y-coordinate or R in terms of a.
b) Find the value of a if the gradient of PR is -2.

Thanks guys.

[Homer]

So If we make y the subject we get y=2-(2/3)x
Therefore p=(3,0) r=(0,2)

Using these values and the formula y-y1=m(x-x1), where m=1/2 you will get the answer to a)

For b) do the same thing and you should end up with two equations, from there you  can easily find the value of a

[Planck's constant]

So If we make y the subject we get y=2-(2/3)x
Therefore p=(3,0) r=(0,2)

Using these values and the formula y-y1=m(x-x1), where m=1/2 you will get the answer to a)

For b) do the same thing and you should end up with two equations, from there you  can easily find the value of a

How does the saying go?
Mind your P's and Q's.

edit: method is good

[Jenny_2108]

Have a question troubling me.

P and Q are the points of intersection of the line y/2 + x/3 =1 with the x and y axes respectively. The gradient of QR is 1/2, where R is a point with x-coordinate 2a, a>0.

a) Find the y-coordinate or R in terms of a.
b) Find the value of a if the gradient of PR is -2.

Thanks guys.

a) Given:

P and Q are the points of intersection of the line y/2 + x/3 =1 with the x and y axes respectively
Thus, x-intercept (y=0) => P (3,0) and y intercept (x=0) => Q (0,2)

Let R(2a, y)



Therefore, y-coordinate of R = a+2

b) P (3,0) and R (2a, a+2)



Havent studied for nearly 2 months so if I do something wrong, pls correct me :p

[jimmy22]

Thanks guys.