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Additional Maths – Trigonometry Functions and Equations, Proofing Trigo Identities

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For my sec 3 Amaths students,

Here are the trigonometry revision questions,

Before you attempt trigo questions, please study and remember the addition angle, double angle and trigo factor formulaes.

Speaking with not a bit of sarcasm tone**   🙂

“Seriously, there is no point doing and practicing the trigo questions without memorising the formulaes. You can do those simple practices for the start but you will face a lot of ‘stuck’ moments, with you trying to flip through the formulae page and the question page when you are solving the more challenging problems.”

Totally ineffective and waste of time. 😦

If you are still unsure on the R-formulae, here are some notes.

For the R-formulae,

It is actually from the addition formulaes.

R cos (x+α)
R [cosxcosα-sinxsinα] ( Think Addition Formula)
(R cosα)cosx-(R sinα)sinx

Hence, if we need to express 5 cos x – 7 sin x in the form of R formulae. (i.e R cos (x+α),

We just compare with the (R cosα) with 5 and (R sinα) with 7. (See that?)

tan α = 7/5 ⇒ α = 60.5°

To find R, we square everything.
R2cos2α+R2sin2α = 52+72
R2[cos2α+sin2α] = 74 ⇒ R =

∴ 5 cos x – 7 sin x =  cos(x+60.5°)

EXAM TIPS:

  • Re-arrange the sine and cosine terms depending on how the question wants you to express the R function.

x sin θ + y cos θ → if you require R sin (θ + α)

  • For R-function, always ensure first term is positive!

  • Double angle formulaes can be applied for all values of x, i.e 4x, 5x, 6x, 7x etc. Not only for the even number coefficients ones.

 cos 7x = cos (2 ( 7/2  ))

  • Only use Double Angle formulae when you can express whole expression in terms of one trigo function (i.e sin, cos, tan)

     

Now, ready?

Try these 5 questions. 🙂

1)                  Solve the following equations for O≤x≤3600

(a)   4 tan x + 2 tan (x-450)  –>Hint: Addition formulae

(b)   5 cos x – 2 sin ( x/2  ) =2 –>Hint: Half Angle formulae and work out the range for the half angle

(c)    cos 8x – 3 cos 4x = -1 –>Hint: Double Angle formulae and work out the range for the double angle identified.

(d)   4 cos x – 3 sin x = 4 –>Hint: R function formulae

(e)   2sin 2x = 7cos2x -2  —>Hint: Not Double Angle formulae. Only use Double Angle formulae when you can express whole expression in terms of one trigo function (i.e sin, cos, tan)

 

Well, another FAMOUS portion of AMaths Trigonometry is proofing identifies.

From what I observed from my previous students, those who are familiar with the trigonometry formulaes, did well naturally. It is actually quite thrilling for these students because they know they definitely got the marks in their pockets when they are able to proof LHS=RHS! woot!

2)                  Prove the following identities

(a)   tan A + cot A = 2cosec 2A

(b)   cosec 2B – cot 2B = tan B

(c)    (tan C +cot C)sin2C =2

(d)   cos D cos 2D – sin D sin 4D = cos 2D cos 3D

(e)  (1+sinE)/(1-sinE)  = (tan E+ sec E)2

Solutions will be posted next week if we did not manage to go through all of them in class. 🙂

Another good news from my upper secondary girl.

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I’m always happy to receive such sms from my students.

She is a very intelligent girl but sometimes she will doubt herself and tell me all the people in her class are so smart. She forgot she is in the same class. 🙂

image

She did not do well for her e maths this round.  I hope she is not demotivated but thrive for better results this EOY exams! 🙂