Mathematics: A Complete Guide to the Course!

[jamonwindeyer]

Hello everyone! The following are all the guides I have written for Mathematics linked in one post, enjoy!

Basic Arithmetic and Algebra

Sequences and Series

Geometry

Plane Geometry

Trigonometry

Functions

Differentiation

Integration

Probability

[brenden]

you'd have to be crazy to not ask questions before half-yearlies! if you don't have an account, just register here, and your questions will get answered within a matter of minutes

[LC14199]

Question, has anyone noticed the bugs in the code for the representation in the differentiation topic? A few lines are showing as the source code for what appears to be a plugin, instead of showing what they should actually show. There are also a number of guides where it looks like there should be something underneath a heading, but there's not, and it actually ends up leaving me more confused because it appears that there's missing information.

Cheers,
LC

[jamonwindeyer]

Question, has anyone noticed the bugs in the code for the representation in the differentiation topic? A few lines are showing as the source code for what appears to be a plugin, instead of showing what they should actually show. There are also a number of guides where it looks like there should be something underneath a heading, but there's not, and it actually ends up leaving me more confused because it appears that there's missing information.

Cheers,
LC

Hey LC! The bugs are not just you, I'm working on fixing them, but it is a tedious process. Any guides in particular you needed fixed ASAP? I'll move them to the top of the list for you!

[LC14199]

Fixing differentiation would be awesome, I have a half yearly coming up for mathematics which encompasses the majority of calculus, and also Locus and the Parabola, along with the basic stuff (Geometry, Linear functions, surds, etc)

[jamonwindeyer]

Fixing differentiation would be awesome, I have a half yearly coming up for mathematics which encompasses the majority of calculus, and also Locus and the Parabola, along with the basic stuff (Geometry, Linear functions, surds, etc)

The Locus/Plane Geometry, Differentiation, and Basic Arithmetic and Algebra guides are all fixed for you! I'll continue to work on getting them all fixed, but as I said, it may take some time, sorry for the inconvenience 

[anotherworld2b]

Hi I was wondering if I could get help breaking down these key points in the syllabus for the wace curriculum for yr11. I have written some things that I assume the key points refer to. However, there are some points that I don't understand.

Cosine and sine rules
1.2.1 review sine, cosine and tangent as ratios of side lengths in right-angled triangles
I assume this refers to memorizing exact values

1.2.2 understand the unit circle definition of cos𝜃𝜃, sin𝜃𝜃 and tan𝜃𝜃 and periodicity using degrees
I am not sure what this point is really talking about

1.2.3 examine the relationship between the angle of inclination of a line and the gradient of that line
Refers to what the angle of elevation is? how to find linear function?

1.2.4 establish and use the cosine and sine rules, including consideration of the ambiguous case and the formula 𝐴𝐴𝐴 𝐴𝐴𝐴 = 12𝑏𝑏𝑏 𝑠𝑠𝑠𝑠𝑠 𝐴𝐴 for the area of a triangle
basically how to use cosine and sine rules? Understanding how to identify an ambiguous case and how to use the formula for the area of a triangle?

Circular measure and radian measure
1.2.5 define and use radian measure and understand its relationship with degree measure
1 radian = 57.3 degrees. How to convert between degrees and radians?

1.2.6 calculate lengths of arcs and areas of sectors and segments in circles
Trigonometric functions
Understanding how to calculate lengths of arcs and areas of sectors and segments in circles Trigonometric functions

1.2.7 understand the unit circle definition of sin𝜃𝜃, cos𝜃𝜃 and tan𝜃𝜃 and periodicity using radians
how the unit circle works?

1.2.8 recognise the exact values of sin𝜃𝜃, cos𝜃𝜃 and tan𝜃𝜃 at integer multiples of 𝜋/6 and 𝜋/4
Memorize exact values

1.2.9 recognise the graphs of 𝑦=sin𝑥,𝑦=cos𝑥, and 𝑦=tan𝑥 on extended domains
not sure what it means by extended domains

1.2.10 examine amplitude changes and the graphs of 𝑦=𝑎sin𝑥 and 𝑦=𝑎cos𝑥
1.2.11 examine period changes and the graphs of 𝑦=sin𝑏 , 𝑦=cos𝑏 and 𝑦=tan𝑏
1.2.12 examine phase changes and the graphs of 𝑦=sin(𝑥−𝑐), 𝑦=cos(𝑥−𝑐) and 𝑦=tan (𝑥−𝑐)
Understanding the different aspects of trig graphs

1.2.13 examine the relationships sin𝑥+𝜋/2=cos𝑥 and cos𝑥−𝜋/2􁉁=sin𝑥
?

1.2.14 prove and apply the angle sum and difference identities
Understanding and applying the identities

1.2.15 identify contexts suitable for modelling by trigonometric functions and use them to solve practical problems
Understanding and applying appropriate trigonmetric functions to solve practical problems?

[RuiAce]

1: That just means sin(theta) = opposite/hypotenuse

2. For that one (which is a bit dodgier), you're basically referring to this diagram. Ignore the stuff about versin and exsec.

3. y=mx+b
m=tan(theta)

4. Basically what you said

5. pi radians = 180 degrees

6. The formula for the area of a sector is \(A=\frac12r^2\theta\) where theta is in radians
For a minor segment it's \(A=\frac12r^2(\theta-\sin\theta)\)

7. Same as 1.2.2, just with radians and not degrees

8. There are tricks for memorising exact values between 0 and pi/2. If you want I can provide that later. The dot point is self explanatory though

9. I think that just means "any" domain. 'Extended domain' is not a term in the world of mathematics. Important thing is you can recognise the graph.

For 10-12, I recommend you experiment using GeoGebra. Just submit the graphs in and see what happens. (Desmos also works)

13. I think you forgot bracketing. But basically \(\sin \left(x+\frac\pi2\right)=\cos x \) is an identity and if you graph it, you should be able to see why.

14. There are tons of identities. Idk which are in your curriculum.

15. Practical problems = you're given a scenario and you have to do it. It's like some of the angle of elevation questions and those really wordy problems you've asked Jamon

[anotherworld2b]

thank you for your help. What kind of tricks are there for memorising exact values between 0 and pi/2?

1: That just means sin(theta) = opposite/hypotenuse

2. For that one (which is a bit dodgier), you're basically referring to this diagram. Ignore the stuff about versin and exsec.

3. y=mx+b
m=tan(theta)

4. Basically what you said

5. pi radians = 180 degrees

6. The formula for the area of a sector is \(A=\frac12r^2\theta\) where theta is in radians
For a minor segment it's \(A=\frac12r^2(\theta-\sin\theta)\)

7. Same as 1.2.2, just with radians and not degrees

8. There are tricks for memorising exact values between 0 and pi/2. If you want I can provide that later. The dot point is self explanatory though

9. I think that just means "any" domain. 'Extended domain' is not a term in the world of mathematics. Important thing is you can recognise the graph.

For 10-12, I recommend you experiment using GeoGebra. Just submit the graphs in and see what happens. (Desmos also works)

13. I think you forgot bracketing. But basically \(\sin \left(x+\frac\pi2\right)=\cos x \) is an identity and if you graph it, you should be able to see why.

14. There are tons of identities. Idk which are in your curriculum.

15. Practical problems = you're given a scenario and you have to do it. It's like some of the angle of elevation questions and those really wordy problems you've asked Jamon

[sweetiepi]

thank you for your help. What kind of tricks are there for memorising exact values between 0 and pi/2?

Re: exact values (sorry for butting in )
(I apologise in advance for the lack of latex haha)
Personally, to remember them I used flashcards to memorise them all until they became automatic.
However some others I know used the 30/60  (pi/3, pi/6) degree and 45 (pi/4) degree triangles method (see below) and others used a really funky method with their hands (which I never personally used, too convoluted for my liking ).
With 0 and pi/2- the easiest way imo is to picture a unit circle (let me know if you haven't covered this! ), where if the angle is at zero, cos is 1, sin is 0 and tan is 0; as well as if the angle is at pi/2, cos is 0, sin is 1 and tan is undefined (because you obviously can't divide by zero! )(due to the trigonometric identity, sin/cos = tan)

However, it is best to find a way that works for you!

triangle method

[RuiAce]

thank you for your help. What kind of tricks are there for memorising exact values between 0 and pi/2?

Nifty right? Just leave that 2 in the denominator, and then for sine you go up from 0 to 4. Then for cosine, you just go down from 4 to 0.

For tan, I only used what made sense to me. I just memorised tan(π/4) = 1 and then was like
Ok, √3 is always involved when I'm dealing with tan.
The smaller tan(π/6) goes with the smaller 1/√3
The larger tan(π/3) goes with the larger √3

[asd987]

How well do I need to know preliminary trig for HSC, any specific topics in particular?

[jamonwindeyer]

How well do I need to know preliminary trig for HSC, any specific topics in particular?

You need to know all of it, all prelim content is assessable at the HSC level! Particularly, you should be good at using the sine/cosine rule and solving trig equations, they are the most common, but you need to be familiar with everything!