WildTrig1: Why Trig is Hard

The usual trigonometry is overly complicated, inaccurate and logically dubious. This is the first of a series that shows you a better way---rational trigonometry! Rational trigonometry replaces distance and angle with more algebraic notions called quadrance and spread. All of those complicated formulas become much simpler, you don't need a calculator any more, and the theory extends Euclidean geometry to arbitrary fields!

What mathematics teacher or student could ask for more?
Assoc. Prof. N J Wildberger from UNSW is also the creator of the MathFoundations series, the WildLinAlg series, and a more advanced video series on Algebraic Topology. Enjoy!

Category:

Science & Technology

Tags:

• trigonometry
• distance
• angle
• rational
• cosine
• sine
• Wildberger
• mathematics
• UNSW
• quadrance
• spread
• universal
• teacher
• student
• course
• calculators

[IClausius]

So we're supposed to throw away the notion of metric space and Euclidian metric - both based on distance??

[njwildberger]

@IClausius I do not advocate throwing away mathematics. But it is necessary to be precise and careful, and if something doesn't work properly, it needs to be fixed or replaced. The notion of metric space in modern mathematics is over-rated, as well as being improperly defined.

[Junkbox09]

hi norman. just a quick correction that there are in fact more than 6 trigonometric identites: there are the versine/cos, as well as the haversine/cos, and even worse, a coversine/cos and an ex(co)secant... yeah it's a lot worse guys, i am trying to amplify the rational trigonometry argument...

[njwildberger]

Hi Junkbox09 While technically that is true, what is usually meant is that there are 6 main identities. From these six main identities one can, using algebraic manipulations, generated lots of additional secondary relations. Thanks for the comment.

[bottle2lip]

my head hurts

[Pebbe496]

One note: Europeans do not measure angels in grads (400gon = full circle). Only surveyors do that (well, I don't know any, but I've read). The confusion is perhaps because the German word for degree is "Grad".

[ghostly65]

How come you don't mention the law of tangents or law of cotangents?

[didu205]

I'm getting more n more interested about trigonometry proofs, I was amazed seeing examples such as the Heron's formula failure with extreme triangles due to round off.

[absolutemes]

Hi this is really nice and I like The "New Trigonometry"

I think use word like "cheat" is really big Arclength "in circle" you can calculate by angle at in radian time the radius and its really works

Thanks

[blue2dada]

If the degree measure of the angles of your triangle is taken to be exact, and the side opposite the 48 degree angle is taken to be exactly 11.8, I find the measures of the remaining sides to be approximately 10.1 and 13.0.

If I treat the sides as being exact I find the approximate measures of the angles (in degrees) to be 44.5, 75.0, and 60.5.

As such it seems the values of your example are not correct.

[taltor727]

@njwildberger  Hi, I'm really interested in this, but does it have practical applications in Calculus, Linear Algebra, and Differential Equations?

[taltor727]

Trig is not too hard, generally its beautiful and powerful. I'm going to give this rational trig a chance since it seems really cool, and I'll look into the hyperbolic geometry at the same time.

[harvellt]

@hiyaitsbruno The magic of vectors.

[hiyaitsbruno]

Why is Trig Even Useful in Life!!!!! GRRRRR -.-