Mr. Dickson



right78190013176251085200Special Angles: left1974850023559487629800Unit Circle: x2+y2=1Note: r>0 ( r = radius )42549405871400Radians: The angle created when the central angle has an arc length equal to the radius. Conversion factor 180oπ or π180o al=θr ( angles must be in radians. ) sinθ=yr, cosθ=xr , tanθ=yx, cscθ=1sinθ=ry, secθ=1cosθ=rx , cotθ=1tanθ=xySolving equationsFind the solution for that is over a finite interval and also a general solution.Ex 1. Solve 2sinx+1=0 , given 0≤x<2π and general solution. sinx=-12 : the angle x is in quadrant III or IV. The ref?= π6 ?x1=7π6 ?x2=11π6 the general solution is ?x1=7π6+2πn ?x2=11π6+2πn n?IEx 2. Solve 3csc2x-4=0 , general solutions. cscx=±23 : the angle x is in all 4 quadrants. The ref?= π3 ?x1=π3 ?x2=2π3 ?x3=4π3 ?x4=5π3 These would all be within one rotation. Notice that 180o separates the angles in quadrant I and III … quadrant II and IV. The general solution for this would be…. ?x1=π3+πn ?x2=2π3+πn n?IEx 3. Solve tan2xsinx=sinx , general solution. sinxtan2x-1=0 → ∴ sinx=0 and tan2x=1 For sinx=0 x= 0 and x= π ∴ x=πn , n?I For tan2x=1 let A=2x so tanA=1 Write a general solution for A ref?=π4 so ?A1=π4+2πn ?A2=5π4+2πn this is simplified to ?A1=π4+πn Change back to x 2x=π4+πn → x=π8+π2nWhat if Ex 3. was to solve over an interval of -π≤x<2π ? x= -π, x=0, x=π for the sine equation and… x= π8 , x= 5π8 , x= 9π8 , x= 13π8 , x= -3π8 , x= -7π8 for the tangent equation.This means there are 9 solutions for this equation. Ex 4. Solve sin2x=sinx , for 0≤x<2π You need to use an identity here sin2x-sinx=0 → 2sinxcosx-sinx=0 → sinx2cosx-1=0 sinx=0 and cosx=12 x=0 , x=π , x= π3 , and x=5π3The above examples show various levels of difficulty involved with solving equations. There are quadratic trigonometric equations, trigonometric equations with b≠1 , and trigonometric equations that involve identities. Identities and ProofsUse your identity sheet to determine if you are correctly using the identities appropriately. In some questions you will be asked to simplify an expression or determine an appropriate identity. Ex 5. Simplify cosx+cotx1+cscx=cosx+cotx1+cscx×cosx+cosxsinx1+1sinx=cosxsinx+cosxsinxsinx+1sinx=cosx(sinx+1)sinx×sinxsinx+1=cosxRestrictions: In the above example we can say cosx+cotx1+cscx=cosx ; however, this is not true for all values. cscx≠-1 → sinx≠-1→ x≠3π2+2πn , n?I cotx=cosxsinx → sinx≠0 → x≠πn cscx=1sinx → sinx≠0 same as aboveHow to do a Proof ? Unfortunately, there is no straight forward answer. You can not move terms from one side to the other. You can only manipulate one side at a time until both sides are equal. Here are a few suggestions.Pick one side and convert to sine or cosine. ***Look for obvious identities : Pythagorean 1-sin2x → cos2x Look for factoring : 1-sin2x=(1-sinx)(1+sinx)Try something else. If you are getting nowhere, try a different approach. ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download