Posted by admin | Posted in Uncategorized | Posted on 02-08-2004
Tags: calculus, education, math, mathematics, precalculus
Pre Calculus Precalculus
Pre-Calculus, 4 midterm questions? I AM CONFUSED! D: (easy points)?
I'm having the first part of my midterm for PreCalculus tomorrow. My teacher said we would have 4 questions, each one dealing with a different division of Trig'.
1st Question: Graphing (trig)
2nd Question: Identity (trig)
3rd Question: solving (trig)
4th Question: application (trig)
When further questioned, she would expand upon what she already told us.. Does anyone have any links or examples of each problem? 
1. graph y = 5 cos(3x + 2) - 4
you need to know 5 is the amplitude, -4 is the vertical shift, -2/3 is the phase shift (2/3 left), and 2π/3 is the wavelength. so you make faint dotted horizontal lines at y = -4 + 5 = 1 (high), at y = -4 (midline), and at y = -4 - 5 = -9 (low). you know the graph looks like a cross between a U and a V; it starts high, goes to mid, then low, then mid, then high. so 1st high point is (-2/3, 1). 1st mid is (-2/3 + π/6, -4), low is (-2/3 + π/3, -9), 2nd mid is (-2/3 + π/2, -4), 2nd high is (-2/3 + 2π/3, 1).
2. prove (sec² Θ - 1) / sec² Θ = sin² Θ.
the thing to remember it's NOT AN EQUATION until you prove it is, so you can't add stuff to both sides, you can't multiply or divide both sides by anything. you use simple identities you already know (definitions, even) and the algebra of fractions, so
(sec² Θ - 1) / sec² Θ = .............. distribute the division
1 - 1/sec² Θ = ........................... use def of secant
1 - cos² Θ = .............................. use basic identity sin² + cos² = 1
1 - (1 - sin² Θ) .......................... and a little simple algebra
sin² Θ
3. find all solutions of cos 2x = 1 + sin² 2x
you'll have to use identities to make it just 1 trig function
cos 2x = 1 + (1 - cos² 2x).............. put the quadratic into std form
cos² 2x + cos 2x - 2 = 0 ............... factor
(cos 2x - 1)(cos 2x + 2) = 0
cos 2x = 1 or cos 2x = -2 ............. 2nd factor has no solution
cos 2x = 1
2x = 0 + 2πn ............................... cos periodic, so add multples of 2π
x = 0 + πn ................................... both terms get divided (shorter wavelength = more values repeated)
4. Out the window of your airplane, which is flying at an altitude of 5 miles, you see 2 towns directly to your left, one at an angle of depression of 28°, the other at an angle of depression of 55°. How far apart are the towns.
10/d1 = tan 55 and 10/d2 = tan 28, so
d2 - d1 = 10/tan 28 - 10/tan 55
If you email me I can send you a copy of the 1st semester precalc final I used to use, just to give you an idea. Details depend on what's been covered, of course.
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Summation Notation