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Maths C3 - Trigonometry... Help??

So I've decided to make a thread for when I'm working through Chapter 6: Trigonometry of the Edexcel C3 Modular Maths Textbook so I can keep posting questions I need help on here instead of posting a new thread every time. I'm self-taught Maths so you will need to be patient!

My first problem I have encountered whilst eating lunch is this...


Q) Work out the exact values (in surd form) of sec(210)

The textbook shows in the example that a Quadrant/Cast Diagram has been used, but is different to the way I've learnt/used it for C2. I thought... sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta} which means since cos is used it should appear in the 4th quadrant of the diagram? I'm so confused as to why it's been used this way :frown:

C3 Chapt.6 EXA2 Q(a).png

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Original post by Philip-flop
So I've decided to make a thread for when I'm working through Chapter 6: Trigonometry of the Edexcel C3 Modular Maths Textbook so I can keep posting questions I need help on here instead of posting a new thread every time. I'm self-taught Maths so you will need to be patient!

My first problem I have encountered whilst eating lunch is this...


Q) Work out the exact values (in surd form) of sec(210)

The textbook shows in the example that a Quadrant/Cast Diagram has been used, but is different to the way I've learnt/used it for C2. I thought... sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta} which means since cos is used it should appear in the 4th quadrant of the diagram? I'm so confused as to why it's been used this way :frown:

C3 Chapt.6 EXA2 Q(a).png


I've never really bothered to learn the quadrant diagram - you can solve all the problems without it
remember sec(x) = 1/cos(x)
that cos(x)=-cos(180+x) (easily derived from cos graph)
and that cos(30)=root(3)/2
and you should be able to solve it
writing C in the 4th quadrant just means that Cos is positive in the 4th quadrant. Angles in all the quadrants have Cosines.

cos 210° will be negative cos ( :teehee: ) 210° is in the 3rd quadrant.*
Eating lunch while doing maths has been scientifically been proven to lead to confusion. Don't do it. :smile:
Reply 4
Avatar for Notnek
Notnek
21
Original post by Philip-flop
So I've decided to make a thread for when I'm working through Chapter 6: Trigonometry of the Edexcel C3 Modular Maths Textbook so I can keep posting questions I need help on here instead of posting a new thread every time. I'm self-taught Maths so you will need to be patient!

My first problem I have encountered whilst eating lunch is this...


Q) Work out the exact values (in surd form) of sec(210)

The textbook shows in the example that a Quadrant/Cast Diagram has been used, but is different to the way I've learnt/used it for C2. I thought... sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta} which means since cos is used it should appear in the 4th quadrant of the diagram? I'm so confused as to why it's been used this way :frown:


Ignoring signs (plus or minus), sin/cos/tan are equal in each of the four quadrants. This is shown below if you draw an X in the CAST diagram:



E.g. let's say the acute angle is 30. This means that ignoring signs:

sin⁡(30)=sin⁡(180−30)=sin⁡(180+30)=sin⁡(360−30)\displaystyle \sin(30) = \sin(180-30) = \sin(180 + 30) = \sin(360-30).

These are the four angles represented in the X of the cast diagram above if you travel anti-clockwise.


And since sin⁡(30)=12\sin(30) = \frac{1}{2}, ignoring signs sin⁡(30)=sin⁡(150)=sin⁡(210)=sin⁡(330)=12\sin(30) = \sin(150) = \sin(210) = \sin(330) = \frac{1}{2}.

The letters A, C, T, S tell you whether it will be 12\frac{1}{2} or −12-\frac{1}{2}

E.g. 210 is an angle in the 'T' quadrant which means that only tan is positive so sin must be negative. So sin⁡(210)=−12\displaystyle \sin(210) = -\frac{1}{2}.


The example you posted has done a similar thing to find cos⁡(210)\cos(210).

For people who struggle with the CAST diagram, I recommend drawing the X and thinking about the four angles.
Original post by notnek
Ignoring signs (plus or minus), sin/cos/tan are equal in each of the four quadrants. This is shown below if you draw an X in the CAST diagram:



E.g. let's say the acute angle is 30. This means that ignoring signs:

sin⁡(30)=sin⁡(180−30)=sin⁡(180+30)=sin⁡(360−30)\displaystyle \sin(30) = \sin(180-30) = \sin(180 + 30) = \sin(360-30).

These are the four angles represented in the X of the cast diagram above if you travel anti-clockwise.


And since sin⁡(30)=12\sin(30) = \frac{1}{2}, ignoring signs sin⁡(30)=sin⁡(150)=sin⁡(210)=sin⁡(330)=12\sin(30) = \sin(150) = \sin(210) = \sin(330) = \frac{1}{2}.

The letters A, C, T, S tell you whether it will be 12\frac{1}{2} or −12-\frac{1}{2}

E.g. 210 is an angle in the 'T' quadrant which means that only tan is positive so sin must be negative. So sin⁡(210)=−12\displaystyle \sin(210) = -\frac{1}{2}.


The example you posted has done a similar thing to find cos⁡(210)\cos(210).

For people who struggle with the CAST diagram, I recommend drawing the X and thinking about the four angles.


Wow, that was a brilliant explanation!! You really cleared that up for me :smile:

So for example... cosine would be positive in the c quadrant but negative in a,s, and t?
Original post by RDKGames
Eating lunch while doing maths has been scientifically been proven to lead to confusion. Don't do it. :smile:


But I have to study every chance I get :frown:
Reply 7
Avatar for Zacken
Zacken
22
Original post by Philip-flop
So for example... cosine would be positive in the c quadrant but negative in a,s, and t?


Not quite!

Cosine is positive in C and A(ll).

So basically the reason why the quadrants are named CAST is for cosine, all, sine, tangent.

Everything is positive in the "all" or "A" quadrant.

Sine is positive in the "sine" or "S" and "all" or "A" quadrants.

Cosine is positive in the "cosine" or "C" and "all" or "A" quadrants.

Tangent is positive in the "tangent" or "R" and "all" or "A" quadrants.

This is just a helpful way to remember what's what, but I do suggest re-reading Notneks post and making sure you understand everything instead of relying on memorisation techniques as above.
Original post by Zacken
Not quite!

Cosine is positive in C and A(ll).

So basically the reason why the quadrants are named CAST is for cosine, all, sine, tangent.

Everything is positive in the "all" or "A" quadrant.

Sine is positive in the "sine" or "S" and "all" or "A" quadrants.

Cosine is positive in the "cosine" or "C" and "all" or "A" quadrants.

Tangent is positive in the "tangent" or "R" and "all" or "A" quadrants.

This is just a helpful way to remember what's what, but I do suggest re-reading Notneks post and making sure you understand everything instead of relying on memorisation techniques as above.


Oh yeah of course!! I am definitely a bit rusty when in comes to CAST diagrams! I'm still trying to understand why things are the way they are instead of just remembering. But as you can see I'm struggling :frown:

I can follow what notnek is saying but I can't completely understand why things are. Like, putting quadrant diagrams aside, I struggle to understand what sin, cos, and tan really is. I know their layout on a stereotypical graph between -360<x<360 but other than typing them into a calculator I'm not sure what their purpose is. cos and sin just look like sound waves to me.
Silly question but is...

A the 1st quadrant
S the 2nd quadrant
T the 3rd quadrant
C the 4th quadrant

on a cast diagram?
Reply 10
Avatar for Zacken
Zacken
22
Original post by Philip-flop
Silly question but is...

A the 1st quadrant
S the 2nd quadrant
T the 3rd quadrant
C the 4th quadrant

on a cast diagram?


Yep.
Can someone please tell me what this means?...

n∈\in Z\mathbb{Z}

I know that x∈Rx\in\mathbb{R} << means that x is any real number. But I don't know what the bit above means :frown:
(edited 7 years ago)
Original post by Philip-flop
Can someone please tell me what this means?...

n∈\in Z\mathbb{Z}

I know that x∈Zx\in\mathbb{Z} << means that x is any real number. But I don't know what the bit above means :frown:


it means n is an integer (n is an element of z (the integer set))
and the second one means that x is an integer, not any real number
(edited 7 years ago)
Original post by Philip-flop
Can someone please tell me what this means?...

n∈\in Z\mathbb{Z}

I know that x∈Zx\in\mathbb{Z} << means that x is any real number. But I don't know what the bit above means :frown:


Don't get confused.

http://www.math.ku.edu/~porter/Math_symbols%20.pdf
Sorry guys I made a typo.. I meant

What does this mean?...
n∈\inZ\mathbb{Z}

I know that x∈Rx\in\mathbb{R} << means that x is any real number
Original post by RDKGames


Wow thanks for that link! Super useful!! :smile:
Hey, in your C3 book, there's a small section called "Lists of symbols and notation". This section is just before the Answers so I'd advise you to use it if you ever need it!
Original post by ManLike007
Hey, in your C3 book, there's a small section called "Lists of symbols and notation". This section is just before the Answers so I'd advise you to use it if you ever need it!


Thank you, I never realised that section in the book before!!
Can someone please explain to me using Trig identities how this...
sec⁡θ=−2.5\sec \theta = -2.5

becomes...

cos⁡θ=−0.4\cos \theta =-0.4

I'm so confused. I know that sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}
which can be rearranged to make cos⁡θ=1sec⁡θ\cos \theta = \frac{1}{\sec \theta}
Original post by Philip-flop
Can someone please explain to me using Trig identities how this...
sec⁡θ=−2.5\sec \theta = -2.5

becomes...

cos⁡θ=−0.4\cos \theta =-0.4

I'm so confused. I know that sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}
which can be rearranged to make cos⁡θ=1sec⁡θ\cos \theta = \frac{1}{\sec \theta}


sec(theta)=-2.5
cos(theta)=1/sec(theta)=1/(-2.5)=-0.4

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