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Unformatted text preview: Math 113, FINAL EXAM, DecemberIIS, 2003. (2 hours) SECTION: Instructor: II. For the angle 6 shown on the ﬁgure: points) 1) Give the exact values of Sr c036 = 5m6 : tan6 = 2) Give the exact value of COS(6 — 3) Give the exact value of 008(26 — III. Given that the circle has a radius of length 5 (units). Evaluate: The length of the arc of circle form A to B, the area of the triangle OBC' and the area of the shaded region. (25
points) ‘ IV. (25 points) 1) Draw the graph of y : 2sin(3:c + — 3. 2) If the graph below is the graph of y = Acos(Ba:+C)+D, with A and B > 0. Among
the quantities A, B, C, D, which ones can be determined from the partial information that
is given? Evaluate them. (25 points) Determine the angle 0 and the mass W so thatthe following is an equilibrium. VI. (This problem must be done without calculator. But you may of course ‘secretely’ use
your calculator to see whether you missed some easy simpliﬁcation.) (20 points) Find the exact value of the cosine of the angle 6 between the two vectors: (7’ =<4,x/§> , l7=<3\/§,7>. What is 6 in radians and in degrees? VII. Complex numbers. (25 points) 1) Let 21 =——1—z'andzg=1—z'.
Evaluate 21 + 22, 2122 and i (in rectangular form a + bi). 2) Show the graphical representation of the complex number 21 = —1 ~ 2', and write 21 in
trigonometric form ( r(cos 9 + isin 3) Find a square root (that you may denote by s) of (—1 —— in trigonometric form. Then,
use the half angle formulas for writing that square root in rectangular form. 4) What are all the square roots of (—1 — i), in rectangular form? VIII. (25 points)
1) Find an angle 51:0 such that sin(m0) = — sin($) 2) Solve sin(w0) : —sin(§), for 0 g a: S 47r. 3) Solve sin2(xo) = Sin2(%)7 for 0 g :1: S 47r. IX. If cos 6 = % and 1275 S 0 g 0, which one of the following angles is equal to cos“1(%1—):
0, —6,7r——0,7r+6?
A short but complete justiﬁcation must be given. (5 points) ...
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 Spring '07
 ROSAY
 Trigonometry, Complex number, Imaginary unit, Euler's formula, exact value

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