Graphing (Method for sin\cos, cos example given)Graphing (Method for cot)Graphing (sec\csc, use previous cosgraph from example above)Tangent Graphing, blast from the pastComposite functions f(f -1(x))Graphing (Method for tan)Write sin\cos equation given graphBasic Sum\Difference formula usages.(Using cos)Basic Half Angle formula usages.Equation (Not so basic)Identities (2)14Equation (Basic)Identities (1)First timers, the buttons above take you to a topic. The home button brings you back. If you find a mistake, IM me at kimtroymath or e-mail me at email@example.com There are MANY things on this test, its a big one. This powerpoint does NOT cover everything. Wait for the review sheet before the test for info, and use this powerpoint to help you cover some of the materials. MATERIAL FOR CH 6 TEST IN RED
A) Factor out the coefficient of x, and use even-odd properties to simplifyFind Amplitude and periodFind Phase Shift, and vertical shiftFind starting and ending x-coordinatesDivide into 4 equal partsLabel key points6) ConnectRemember, cos(x) = cos(-x)You will always do this, this is part of your work on a test and is requiredAmplitude =
V.S. =Starting point is phase shift. Ending point is Phase shift + Period35+You will take the starting and ending points and find the average, then find the average again to break it up into four equal regionsYou want to study the sine and cosine graphs. Remember:Sine 0, 1, 0, -1, 0Cosine 1, 0, -1, 0, 1You are basically performing transformations on those key points
A) Factor out the coefficient of x, and use even-odd properties to simplifyFind Vertical Stretch and periodFind Phase Shift, and vertical shiftFind starting and ending x-coordinatesDivide into 4 equal partsLabel key points6) ConnectVertical = StretchPeriod =
Phase Shift =
Vertical Shift = Find the average, then find the averages again.cot, asymp 1 0 -1 asymp
A) Factor out the coefficient of x, and use even-odd properties to simplifyFind Vertical Stretch and periodFind Phase Shift, and vertical shiftShift zero (the middle)Divide the period in half, add and subtract from the middle, sketch asymptotes.Perform transformations.6) ConnectVertical = StretchT =
V.S. =Remember, its \2-1, 1 betweenGo between the asymptotes and the middle and put -1, 1, then transform.
First, sketch the cos graph, stating all the information as your for cosine.At the zeros of cosine (or the middle), sketch asymptotes.At the maxes and mins (tops and bottoms), make your Us35You arent going to need three cycles. Probably just one cycle.
Write the equation of the sin and cos graph.Remember, for sin and cos, the amplitude, period, and vertical shift are all the same, only the phase shift is different.Amplitude. You can use common sense, how far is the max from the middle, or you can use the formula.Vertical Shift, how much did the middle move from the x-axis? Or you can use the formula.ClearVertical ShiftAmplitudeClearPeriodClearcos Phase ShiftClearsin Phase ShiftClearConcept, V.S. is average of max and min. So you add and divide by 2.Concept, Amplitude is the distance between max and min, so you subtract (distance) then divide by 2.The period of sin and cos are the same, so its easiest (IMHO) to find the period using cosine. To do that, find out how far apart the maxes are.Cosine starts at the top (1 0 -1 0 1). So find a max, the x-coordinate is a possible phase shift.There are many possible solutions. For this problem, you could use -3pi\4, pi\4, 5pi\4, etc.Sin starts in the middle, then goes up (0 1 0 -1 0). Find a point in the middle where the graph goes up afterwards. That is one possible phase shift.-pi, 0, and pi are all viable options.
You could try to graph tangent using old style transformations if you wanted to.Factor out the coefficient of x.HorizontalStretch:Reflect:Shift:VerticalStretch:Reflect:Shift:Asymptote changes are only affected by horizontal transformations.Key PointsI recommend the other method for faster graphing.This is different than the other tangent graph because there is an addition sign inside, not a subtraction sign.
Composite trig functions.Set up triangle in the correct quadrant.Pythagorean Theorem may be necessary. (r always positive)Find solution using correct sides.Note, secant negative means its in quadrant II.Remember, r is positive.
Basic sum\difference formula usages.Note, only showing examples for sin. These problems may show up in cos and tan format. If you need to do cot, sec, or csc, use the formula of their reciprocal counterparts, then take the reciprocal of the solution. You need to be careful though, you will have to rationalize the denominator at times. The CONJUGATE will be very helpful.
In this example, to find csc, I use sin to find a solution, and since csc is the reciprocal of sin, I take the reciprocal of my answer. The work is shown on how to break it down. Watch for the conjugate.ApplicationClearReverseClearNoteClearBreak it up into common radian values. A chart may be helpful. (Note, use same denominators)Many combinations are possible. These add up to equal 5pi over 12.Apply appropriate formulaMatch up the expression with the correct formula.These problems are designed to give you a familiar unit circle value.
Basic Half-Angle formula UsageWhat quadrant is A\2 in?Set up triangle, use original A when plugging into formula.Your solution will be plus OR minus, not plus and minus. Use the quadrant of A\2 to determine SIGN!Cosine is negative in quadrant II, so we will use the negative sign.AThese may also be done with sum\difference formulas (45o 30o), but see if doubling it may give you a common unit circle measure.Look at 15o, its in quadrant I, so sin is POSITIVE in quadrant I.These problems will also involve radians. They work in a similar fashion.
Equations (Basic)Give the general formula for all the solutions.Be careful, sometimes you can combine general formulas.
This does NOT cover all possible types of equations. Some common things to watch out for regarding equations:Move everything to one side.Use properties when possible (Sum to Product, pythagorean, double angle)Many times, changing things into the same trig function may be helpful.Factoring may occur many times.Remember to plus\minus when square rooting both sides.You can combine general formulas sometimes.Give the general formula for all the solutions.
Identities: Pg 513: 45RULE: WORK ON ONE SIDE ONLY!Remember the question mark.Helpful Items, in no particular order.Changing to sin\cos helps.Look for Pythagorean, double angle, product-to-sum, reciprocal, even\odd identities.Look at the other side.Conjugates and multilying by one helps.Combining or splitting up fractions is also helpful.Factoring may be helpful.Work on more complicated side.I noticed there was cot on the other side, thats why I didnt change to sin\cos in this case.
Identities: Pg 512: 30RULE: WORK ON ONE SIDE ONLY!Remember the question mark.Helpful Items, in no particular order.Changing to sin\cos helps.Look for Pythagorean, double angle, product-to-sum, reciprocal, even\odd identities.Look at the other side.Conjugates and multilying by one helps.Combining or splitting up fractions is also helpful.Factoring may be helpful.Work on more complicated side.Here is another method