Graphing (Method for sin\cos, cos example given)

1. Graphing (Method for sin\cos, cos example given) Basic Sum\Difference formula usages. (Using cos) Graphing (Method for cot) Basic Half Angle formula usages. Graphing (Method for tan) Equation (Basic) Graphing (sec\csc, use previous cos graph from example above) Equation (Not so basic) Write sin\cos equation given graph Identities (1) Tangent Graphing, blast from the past Identities (2) 14 Composite functions f(f -1(x)) 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 sakim@fjuhsd.net There are MANY things on this test, it’s 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

2. A) Factor out the coefficient of x, and use even-odd properties to simplify • Find Amplitude and period • Find Phase Shift, and vertical shift • Find starting and ending x-coordinates • Divide into 4 equal parts • Label key points • 6) Connect 1 5 3 Amplitude = T = P.S. = V.S. = Remember, cos(x) = cos(-x) You will always do this, this is part of your ‘work’ on a test and is required + You want to study the sine and cosine graphs. Remember: Sine  0, 1, 0, -1, 0 Cosine  1, 0, -1, 0, 1 You are basically performing transformations on those key points Starting point is phase shift. Ending point is Phase shift + Period You will take the starting and ending points and find the average, then find the average again to break it up into four equal regions

3. A) Factor out the coefficient of x, and use even-odd properties to simplify • Find Vertical Stretch and period • Find Phase Shift, and vertical shift • Find starting and ending x-coordinates • Divide into 4 equal parts • Label key points • 6) Connect END + START START END cot, asymp 1 0 -1 asymp Find the average, then find the averages again. Vertical = Stretch Period = Phase Shift = Vertical Shift =

4. A) Factor out the coefficient of x, and use even-odd properties to simplify • Find Vertical Stretch and period • Find Phase Shift, and vertical shift • Shift zero (the middle) • Divide the period in half, add and subtract from the middle, sketch asymptotes. • Perform transformations. • 6) Connect -1, 1 between Go between the asymptotes and the middle and put -1, 1, then transform. Vertical = Stretch T = P.S. = V.S. = Remember, it’s π\2

5. 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 U’s 1 5 3 You aren’t going to need three cycles. Probably just one cycle.

6. 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. Max Min Max Max middle middle middle Then up Then up Then up Cosine starts at the top (1 0 -1 0 1). So find a max, the x-coordinate is a possible phase shift. Vertical Shift, how much did the middle move from the x-axis? Or you can use the formula. Amplitude. You can use common sense, how far is the max from the middle, or you can use the formula. The period of sin and cos are the same, so it’s easiest (IMHO) to find the period using cosine. To do that, find out how far apart the maxes are. 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. Concept, Amplitude is the distance between max and min, so you subtract (distance) then divide by 2. There are many possible solutions. For this problem, you could use -3pi\4, pi\4, 5pi\4, etc. Concept, V.S. is average of max and min. So you add and divide by 2. -pi, 0, and pi are all viable options. Vertical Shift Amplitude Period cos Phase Shift sin Phase Shift Clear Clear Clear Clear Clear

7. You could try to graph tangent using old style transformations if you wanted to. Factor out the coefficient of x. Horizontal Stretch: Reflect: Shift: Vertical Stretch: Reflect: Shift: Asymptote changes are only affected by horizontal transformations. Key Points I 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.

8. Composite trig functions. • Set up triangle in the correct quadrant. • Pythagorean Theorem may be necessary. (r always positive) • Find solution using correct sides. Remember, r is positive. Note, secant negative means it’s in quadrant II.

9. Basic sum\difference formula usages. Break it up into common radian values. A chart may be helpful. (Note, use same denominators) 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. Match up the expression with the correct formula. These problems are designed to give you a familiar unit circle value. Apply appropriate formula Many combinations are possible. These add up to equal 5pi over 12. Application Reverse Note Clear Clear Clear

10. Basic Half-Angle formula Usage What quadrant is A\2 in? Common denominator 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! A Cosine is negative in quadrant II, so we will use the negative sign. These may also be done with sum\difference formulas (45o – 30o), but see if doubling it may give you a common unit circle measure. These problems will also involve radians. They work in a similar fashion. Look at 15o, it’s in quadrant I, so sin is POSITIVE in quadrant I.

11. Give the general formula for all the solutions. Be careful, sometimes you can combine general formulas. Equations (Basic)

12. Give the general formula for all the solutions. • 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.

13. Identities: Pg 513: 45 • RULE: 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, that’s why I didn’t change to sin\cos in this case.

14. Identities: Pg 512: 30 • RULE: 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

15. 14