Topic 2.1 Trigonometry

Syllabus

Topic 2.1: Trigonometry (16 hours)

The basic trigonometric functions

2.1.1 determine all solutions of f(a(x-b))=c where f is one of sine, cosine or tangent

2.1.2 graph functions with rules of the form y=f(a(x-b))+c where f is one of sine, cosine, or tangent

Compound angles

2.1.3 prove and apply the angle sum, difference, and double angle identities

The reciprocal trigonometric functions, secant, cosecant and cotangent

2.1.4 define the reciprocal trigonometric functions; sketch their graphs and graph simple transformations of them

Trigonometric identities

2.1.5 prove and apply the Pythagorean identities

2.1.6 prove and apply the identities for products of sines and cosines expressed as sums and differences

2.1.7 convert sums acos x +bsin x to Rcos(x+theta) or Rsin(x+theta) and apply these to sketch graphs; solve equations of the form acos x+bsin x=c

2.1.8 prove and apply other trigonometric identities such as cos3x=4cos^3 x-3cos x

Applications of trigonometric functions to model periodic phenomena

2.1.9 model periodic motion using sine and cosine functions and understand the relevance of the period and amplitude of these functions in the model

Lessons

Graphs of y=f(a(x-b))

(Note that this Prezi may not yet have a voiceover. I do intend to add one, but I wanted to get it up as soon as possible, it’s late, and I don’t have much voice.)

Also see Mr Woo’s video on graphing trig functions.

Solutions of f(a(x-b))=c

If you are unsure about any of this, I recommend you view Mr Woo’s series of five videos “Solving Trigonometric Equations”. Even if you feel you have a good handle on this, you should still view the three videos “Harder Trigonometric Equations”.

Compound Angle Identities

Reciprocal Trig Functions

https://prezi.com/view/nlg8eezJSBbJmdTJbhE3/

See Mister Woo’s lesson on trig functions and their reciprocals. This contains a very nice ‘cheat’ to help you remember which function each is the reciprocal of.

See the nice proof by Mister Woo that Tangent and Radius are perpendicular. This is interesting for trigonometry, and is also a lovely example of a proof by contradiction.

See the clip by Mister Woo on the significance of the names ‘tangent’ and ‘secant’.

Pythagorean Identities

There are three main forms for the Pythagorean identity:

  • The principal form: sin^2(x)+cos^2(x)=1
  • Divide the principal form by  cos^2(x) gives  tan^2(x)+1=sec^2(x)
  • Divide the principal form by  sin^2(x) gives  1+cot^2(x)=csc^2(x)

Students should be able to prove the Pythagorean identity, and show how the other forms can be obtained from the principal form. At time of writing this, the Wikipedia article on the Pythagorean Trigonometric Identities has a couple of nice proofs that are at the right level for this class.

Product to Sum and Sum to Product

Convert sums acos x +bsin x to Rcos(x+theta) or Rsin(x+theta) and applications

Proving Other Trigonometric Identities

Modelling Periodic Motion

Other Resources

  • Khan Academy offers many videos that are helpful.
  • Wootube gives videos with a different style based on Mr Eddie Woo’s classroom presentations.
  • Wikipedia is quite reliable and valuable for mathematical topics (although it often goes further than needed for our purposes). For example, see the article on the Pythagorean Trigonometric Identities.

 

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