Finding trig functions given another or a point (i.e. sin θ = 3 5. Finding trig functions given quadrant and line equation (Problems in 6.


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1 1 Math 3 Final Review Guide This is a summary list of many concepts covered in the different sections and some examples of types of problems that may appear on the Final Exam. The list is not exhaustive, but highlights most types of questions that you should be able to answer/solve. Some conceptual questions are included. If you have any questions about what is meant by a certain item in the list, please ask me for clarification. Caveat: This list was written late at night  if you see a typo, let me know so that it can be corrected/clarified for everyone. 6.1 What is an angle? What are radians and degrees? Arc length Area of a sector Finding positive and negative coterminal angles Drawing an angle in standard position Converting between radians and degrees Angular and linear speed 6.2 Finding trig functions of any angle Finding trig functions given another or a point (i.e. sin θ = 3 5 or given point P ( 2, 5)) Finding trig functions given quadrant and line equation (Problems in 6.2) sec 2 θ) 6.3 Use reciprocal identities (e.g. sin θ = 1 csc θ ) and Pythagorean identities (e.g. tan2 θ + 1 = What is the unit circle? Why is it helpful? Find trig functions of an angle given a point Find versions of a point given initial position e.g. P (t) = ( 3, 4, find P ( t + π) 5 5
2 2 Find a point on the unit circle given an angle in standard position Use formulas for negatives, e.g. sin( θ) = sin θ 6.4 What is a reference angle? 6.5, 6.6 Finding reference angles for a given angle What affect do amplitude, phase shift, vertical shifts, vertical/horizontal rotations, and period change have on a graph? Graph the sine, cosine, tangent, cotangent, secant, and cosecant functions with the changes mentioned above Applied problems with graphing Find domain/range/asymptotes of different trig functions Describe behavior at a value or infinitely (e.g. as x π 2 +, tan x?) Use trig functions to solve applied/word problems involving right triangles Prove various trigonometric identities Disprove a stated identity 7.2 Solve trigonometric equations where the solutions are special angles (e.g. 2 sin 3θ+ 2 = 0 or (2 sin x + 1)(2 cos(2x π 3 ) 3) = 0) Find solutions on a given interval Find generic solutions (all solutions) Find solutions and interpret them as part of an applied problem (e.g. Problem number 78 in 7.2) 7.3 and 7.4 tell us? What do the addition/subtraction/complementary/double angle/half angle/etc identities
3 3 Use the addition/subtraction/complementary/double angle/half angle/etc identities in proving identities Use the addition/subtraction/complementary/double angle/half angle/etc identities in solving equations (e.g. Problem number 60 in 7.3 and problem number 41 in 7.4) find the exact value of an expression (e.g. cos 7π; cos(x + π 24 ) if sin x = ; find cos 2θ and sin θ/ if tan θ = 3) 4 interpret a function in terms of graph (e.g. Problem number 65 in 7.3 and problem number 46 in 7.4) reduce a power of an expression (e.g. Problem number 31 in 7.4) 7.6 What does sin 1 x represent? What about for cosine, tangent, cotangent, secant, and cosecant? What is the domain/range of different inverse trig functions? Find inverse trig functions of given ratios (e.g. find sin 1 ( 1 2 )) Evaluate expressions with inverse trig (e.g. tan[sin 1 ( 1 2 )] or sin 1 [sin( 5π 6 )]) Write algebraic expressions for inverse trig expressions (e.g. sin(tan 1 ( 1 x ))) 7.6) Evaluate inverse trig expressions where identities are needed (e.g. Problem number 21 in Solve equations using inverse trig functions (e.g. problem number 60 in 7.6) Use inverse trig in applied problem (e.g. problem number 68 or 69 in 7.6) 8.1 and 8.2 What is the Law of Sines? What is the Law of Cosines? What do the laws say about side/angle ratios? What types of triangles can the laws be used on? Solve for all missing sides and angles in a triangle, using the Law of Sines or Law of Cosines.
4 4 Solve for all missing sides and angles in a triangle where there are two possible triangles (ambiguous case with Law of Sines) Apply the Law of Sines and/or Law of Cosines to solve an applied problem 8.3 What are vectors? How are they used? What is the magnitude of a vector? What is scalar multiplication and what affect does it have on a vector? What are i and j and why are they important? What is a unit vector? Add, subtract, find scalar multiples, and find magnitudes of vectors Prove properties of vectors (e.g. prove a + 0 = a) 2 i 5 j) Find a unit vector given a particular vector (e.g. find a unit vector in the direction of Find a vector in the same direction as a given vector with different magnitude (e.g. find a vector in the direction of 1, 7 with magnitude of 10) Find a vector given its direction and magnitude (e.g. Problem number 40 in 8.3) Use vectors to find resultant force or direction in applied problems (e.g. Problem numbers 47, 63, or 67 in 8.3) 8.5 Find the absolute value of a complex number Plot a complex number on a real/imaginary plane (e.g. represent a number geometrically  such as problem number 17 in 8.5) Express a complex number in trigonometric form for a given restriction on θ (e.g. Express 3 3 3i in trig form with 2π < θ < 4i; Express 2 7i in trig form for 0 < θ < 2π) Expression a given complex number in the form a + bi when given it in trig form (e.g. Problem number 51 in 8.5) 8.6
5 5 Not on Final Exam 11.5 What are polar coordinates? What do they represent? Why are they important? Represent a polar point in more than one way (e.g. Problem number 2 in 11.5; Find a different pair of polar coordinates that represent the same point as (r, θ) = (5, π 8 )) Change polar coordinates to rectangular coordinates Change rectangular coordinates to polar coordinates for given values of θ (e.g. Find polar coordinates for (x, y) = ( 8, 8 3) where 2π < θ < 0) Find a polar equation for a given equation in x, y Find an x, y equation for a given polar equation
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