Trigonometric & Polar Functions unit test
A test on this unit alone, marked as a percentage and a letter grade — for the test your class is actually sitting, rather than for May. Answer everything, then submit once: seeing the answer to question 3 before attempting question 4 makes the final percentage meaningless.
No-triangle case in SSA
Arc length and sector area
Reciprocal functions
Tangent function behavior
Diameter vs radius in r = a cos θ
Period from consecutive extremes
sin⁻¹ is not a reciprocal
arcsin(sin x) is not always x
Building a reciprocal graph from its sinusoid
Stretching the interval for sin(bx)
Quadrant of a doubled angle
Asymptotes of tangent
Short answer 1. Define or explain: Why arcsin is restricted
3 ptsShort answer 2. Define or explain: Law of cosines reduces to Pythagoras
3 ptsShort answer 3. Define or explain: Amplitude and midline
3 ptsShort answer 4. Define or explain: Polar coordinates
3 ptsFree response
6 ptsMODELING A PERIODIC CONTEXT (Question 3, no calculator, 6 points). A vibrating guitar string's motion is modeled by a periodic function. At time t = 0 seconds, point X on the string starts at its highest position, 2 millimeters above its resting position, passes through the resting position to its lowest position 2 mm below, and returns to 2 mm above. This full cycle occurs 200 times per second. The sinusoidal function h models the displacement of X from rest, in millimeters (positive above, negative below), as a function of time t in seconds. The graph of h and its dashed midline are shown for two full cycles, with five labeled points and no scales: F at the initial maximum, G at the next intersection with the midline, J at the following minimum, K at the next midline intersection, and P at the following maximum.
A. Determine possible coordinates (t, h(t)) for the five points F, G, J, K, and P.
B. The function h can be written in the form h(t) = a·sin(b(t + c)) + d. Find values of the constants a, b, c, and d.
C. The t-coordinate of G is t1 and the t-coordinate of J is t2. C(i). On the interval (t1, t2), is h (a) positive and increasing, (b) positive and decreasing, (c) negative and increasing, or (d) negative and decreasing? C(ii). On (t1, t2), describe the concavity of the graph of h and determine whether the rate of change of h is increasing or decreasing.