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Unformatted text preview: Basic Stuff 1.1 Trigonometry The common trigonometric functions are familiar to you, but do you know some of the tricks to remember (or to derive quickly) the common identities among them? Given the sine of an angle, what is its tangent? Given its tangent, what is its cosine? All of these simple but occasionally useful relations can be derived in about two seconds if you understand the idea behind one picture. Suppose for example that you know the tangent of θ , what is sin θ ? Draw a right triangle and designate the tangent of θ as x , so you can draw a triangle with tan θ = x/ 1 . 1 θ x The Pythagorean theorem says that the third side is √ 1 + x 2 . You now read the sine from the triangle as x/ √ 1 + x 2 , so sin θ = tan θ √ 1 + tan 2 θ Any other such relation is done the same way. You know the cosine, so what’s the cotangent? Draw a different triangle where the cosine is x/ 1 . Radians When you take the sine or cosine of an angle, what units do you use? Degrees? Radians? Cycles? And who invented radians? Why is this the unit you see so often in calculus texts? That there are 360 ◦ in a circle is something that you can blame on the Sumerians, but where did this other unit come from? R 2 R s θ 2 θ It results from one figure and the relation between the radius of the circle, the angle drawn, and the length of the arc shown. If you remember the equation s = Rθ , does that mean that for a full circle θ = 360 ◦ so s = 360 R ? No. For some reason this equation is valid only in radians. The reasoning comes down to a couple of observations. You can see from the drawing that s is proportional to θ — double θ and you double s . The same observation holds about the relation between s and R , a direct proportionality. Put these together in a single equation and you can conclude that s = CRθ where C is some constant of proportionality. Now what is C ? You know that the whole circumference of the circle is 2 πR , so if θ = 360 ◦ , then 2 πR = CR 360 ◦ , and C = π 180 degree- 1 It has to have these units so that the left side, s , comes out as a length when the degree units cancel. This is an awkward equation to work with, and it becomes very awkward when you try to do calculus. An increment of one in Δ θ is big if you’re in radians, and small if you’re in degrees, so it should be no surprise that Δsin θ/ Δ θ is much smaller in the latter units: d dθ sin θ = π 180 cos θ in degrees James Nearing, University of Miami 1 1—Basic Stuff 2 This is the reason that the radian was invented. The radian is the unit designed so that the propor- tionality constant is one. C = 1 radian- 1 then s = ( 1 radian- 1 ) Rθ In practice, no one ever writes it this way. It’s the custom simply to omit the C and to say that s = Rθ with θ restricted to radians — it saves a lot of writing. How big is a radian? A full circle has circumference 2 πR , and this equals Rθ when you’ve taken C to be one. It says that the angle for ato be one....
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This note was uploaded on 01/30/2012 for the course PHYS 315 taught by Professor Nearing during the Fall '08 term at University of Miami.
- Fall '08