## Monday, March 17, 2008

### Pythagorean Identities

OUTLINE:
* Discussed a case study, a blogging issue, that is of recent concern in Australia.
* Continued on solving the introductory expressions on trigonometric identites.
* Discovered the concept of corrolories and how it they can make 'algebraically massaging' an expression much easier.

CASE STUDY: AL UPTON AND THE MINILEGENDS ARE SHUT DOWN

The day started off with an appealing discussion on a man named Mr. Al Upton, a sixth grade teacher in Australia, whose blog for his students was recently shut down by order of the Ministry of Education. The reasons behind the shutting down of his educational blog is not specific, but this decision is believed to contradict Article 13 of the Convention on the Rights of the Child in which the child is allowed to use any type of media for her or his education, basically. This alarming issue was discussed by Mr. Kuropatwa to his class as a warning that our blogs could be shut down as well. Mr. K encouraged us to comment on Upton's blog so as to raise the concern even further that Canadian children (us) are concerned about this as well and that children should be able to use blogs for educational purposes.

Now on to today's lesson.

INTRODUCTORY EXPRESSIONS OF TRIGONOMETRIC IDENTITIES

Refer to SLIDES 4 to 7.

We 'algebraically massaged' (sec t)/(cos t) - (sec t)(cos t) and (1 + tan^2 B)/ tan^2 B. But that work is now lost and Mr.K had to recreate them as that is mentioned in his post.

PYTHAGOREAN IDENTITIES

Refer to SLIDE 8.

One can find that any triangle on the circle that the equation...

sin^2 X + cos^2 X = 1

...with the case of degenerate triangles at the corner points on the unit circle (i.e., 0, pi/2, pi, 3pi/2, and 2pi in the domain of [0,2pi]) in which the 'triangle' has an adjacent or opposite side equal to zero, therefore, "missing a hypotenuse" and the 'triangle' can be seen as "degenerate," but still abides to the "sin^2 X + cos^2 X = 1" rule.

DERIVING COROLLARIES FROM sin^2 X + cos^2 X = 1.

Refer to SLIDE 9.

If sin^2 X + cos^2 X = 1, then the following must also be true...

sin^2 X = 1 - cos^2 X
cos^2 X = 1 - sin^2 X

...because they are algebraically the same.

So anytime we can see sin^2 X, cos^2 X, 1 - cos^2 X, and 1 - sin^2 X in doing trigonometric identities questions, we can substitute those in.

For example, refer to SLIDE 10.

From our previous class, we 'algebraically massaged' (1 - cos^2 A)/cos^2 A. We substituted/replaced the 1 - cos^2 A with a sin^2 A and found out that the equation can further be simplified to (sin^2 A)/(cos^2 A) or tan^2 A.

Refer to SLIDE 11.

And if we use that same equation, sin^2 X + cos^2 X = 1, and divided that whole equation by sin^2 X or cos^2 X, we can derive more corrolaries that are algebraically the same as its siblings.

SUMMARY
* Mr. Al Upton and the miniLegends' blog was shut down by the Ministry of Education in Australia because the ministry felt the blog violated some laws. This can happen to us!
* From sin^2 X + cos^2 X = 1, we can derive many more equations--the corrolaries--and use them in 'algebraically massaging' trigonometric identities questions.

HOMEWORK:
Exercise 14: Trigonometric Identities I

Next scribe is Joyce.

m@rk said...

Zeph,

I should say that I'm really impressed by this post. This post is very well written and organized. I like how you make outlines, it sort of summarized everything that happened in class. Keep up the good work!

-m@rk

zeph said...

Thank you.

I appreciate your unexpected comment. =)