MAT123 Fall 2003 Review for Final Exam

Final review should use this sheet along with the review sheets for Midterm 1 and Midterm 2.

6.4 Be able to use the Law of Sines (Box, p.506) to solve triangles when you know one side and two angles: ASA (Example 1, Exercises 2, 3), or SAA (Example 2, Exercises 1, 6). In these two cases all three of the angles in the triangle are known. Understand the ambiguity when you apply the Law of Sines to SSA problems (Examples 3, 4, 5): if you know sinθ there are in general two possible θs between 0° and 180°, and sometimes there are two different solutions to the problem (Exercises 15, 16, 32). The sin-1 key on your calculator only gives you the angle betweenand 90° with that sine. Understand this fact of life.

6.4 Be able to use the Law of Cosines (Box, p.513) to solve triangles when you know all three sides (Example 2, Exercises 5, 6, 8) or two sides and the angle between them: SAS (Example 1, Exercises 1, 2, 3, 27).

5.1 Be comfortable with the unit circle. The circumference is 2π = 6.28.. Be able to estimate locations of angles from their radian measure (Exercises 43-46). Be comfortable in converting radians to degrees, and degrees to radians.

5.2 In your revision, concentrate on the functions sin, cos, and tan, and the fundamental identities tan = sin/cos and sin2+cos2=1. Always remember the definitions of sin and cos in terms of the unit circle (Box, p.418). These definitions should allow you to reconstruct the "even-odd properties" (Box p. 423, first row) and the "reduction formulas" (Exercises 77, 78). You should know the "special values" Table 1, first three columns, p.420 and, in case you forget them, how to derive them from the right isoceles triangle and the equilateral triangle (as in Fig 5 and 6 p.484). When you use your calculator in these sections, make sure it is in radian mode. Exercises 27-34, 35-42.

5.3 sin and cos have period 2 pi: they repeat exactly every 2 pi units. The function y=a sin k(x-b) can be derived from y=sin(x) by three of our elementary changes: sin(x) -> sin(kx) -> sin(k[x-b]) -> a sin(k[x-b]), corresponding to a compression of the graph by a factor of k (so the new period is 2 pi/k), a shift to the right by b units, and a vertical scaling by a. The absolute value |a| is the amplitude, and b is the phase shift. If you are given the function in the form sin(kx-d) be sure to rewrite it as sin(k[x-d/k]) before evaluating the phase shift and using it to sketch the graph. Exercises 21-34.

5.4 Here concentrate on the graph of tan. The function has period pi. Be able to graph functions of the form a tan([k(x-b)] by the same methods as before. Examples 1,2, Exercises 1,4,19,31.

7.1 The best method for dealing with trigonometric identities is to 1) rewrite everything in terms of sin and cos and 2) try to rearrange terms so as to be able to apply sin2+cos2=1. Other algebraic identities you may need are (a+b)2 = a2+2ab+b2 and a2-b2 = (a+b)(a-b). Exercises 10,21

7.2 Do not try to memorize the addition formulas for sin and cos. If you need them on the test they will be written out there. Be able to use the formulas (Examples 1,2,3, Exercises 1-8, 11, 12, 21-24

7.4 Understand the three basic inverse trigonometric functions sin-1, cos-1, tan-1, preferably in terms of their definitions on the unit-circle diagram. Boxes on p.561 and the text in italics, Boxes on p.563 and the text in italics, Boxes on p.564-565 and the text in italics. Be able to compose one trigonometric function with the inverse of another, as in Example 5. Exercises 13-20, 23, 25, 26.

Calculus 2.1 Understand the relation between the tangent (line) to the curve y=f(x) at the point (a,f(a)) and a secant (line) passing through the points (a,f(a)) and (a+h,f(a+h)) for h nonzero. We can calculate the slope of the secant by the "rise over run" method. If the curve is smooth (Fig. 3 p.96) the secants will approximate the tangent as h becomes small. In case x is time and f(x) is position, understand that the slope of the secant is the average velocity between a and a+h. Exercises 4, 5.

Calculus 2.3 and 2.6 Be able to calculate limits of the type encountered in tangent and velocity problems, in case the function f(x) is a polynomial of degree two. Example 3 p.113, Example 5 p.114. Example 1 p.143, Example 3 p.145. Be able to use these limits to calculate the equation of the tangent line by the point-slope method. Exercises 15, 16ab, 18, 20, 21 p.148-149

December 6, 2003