Notes for Homework #13

Notes on parametric equations

Usually, we have a function with two variables, the dependent variable if either y or f(x) and the independent variable is x. In this case, we will instead have an independent variable t and two equations that are dependent on t, which in this case can be thought of as time measured in seconds. 

We will consider the path of a projectile being acted upon by gravity, where H(t) is the height and D(t) is the distance. We need two numbers to define our functions, v0 is the initial velocity - known as muzzle speed - and the angle at which the project is launched, which we will call theta. The two formulas are as follows, measuring time in seconds and height in feet

H(t) = t(sin(theta)v0 - 16t)
D(t) = t(cos(theta)v0)

Height will be 0 at t = 0 and when sin(theta)v0 - 16t = 0, which can be thought of as the launch and the landing, respectively.
When you solve the landing time, plug in that t into D(t) to get how far the projectile will travel before it hits the ground.


Notes on the angle between two vectors

Biographical notes on five computer scientists

Link to the biographies of John Von Neumann, Grace Hopper, Alan Turing, Edsger Dykstra and Donald Knuth

Notes for Homework #12, due Tues. Apr. 30 NOT ACCEPTED LATE

Notes for converting polar to Cartesian and vice versa

Note for plotting polar coordinate functions

The website Wolfram Alpha is a good resource for plotting. For example, use the instruction

plot r = cos(theta)

To see an accurate picture of a circle with its center at (= ½, 0) and radius = ½, passing through both (0, 0) and (1, 0)

Note: The x-axis is all points of the form (x, 0) in Cartesian coordinates. The form looks exactly the same in Polar coordinates, since the angle 0 is the x-axis.

Notes for Homework #11, due Tues., April 23

Notes for conversion from polar coordinates to Cartesian coordinates and vice versa.


 

Notes on Homework 10, due Tuesday, April 16

Changing f(x) = msinx + ncosx into f(x) = Asin(x + c)
 

Notes on Homework #9, due Tuesday, April 9

Notes on how to plot the defining points of f(x) = Asin(bx+c) + D, where the defining points are the start at the median value and the closest max and min.

Notes for Homework #8, due Tues., March 19

Notes on the law of sines and how it differs from the law of cosines.

Notes for Homework 6, due March 5

Notes for the double angle formulae and the addition and subtraction of angles

Notes on trig identities
 

Notes on Homework 5, due Feb. 27

Notes on inverse trig functions

Notes on decimal degrees, degrees-minutes-seconds (DMS), radians and radians given as multiples of pi

Notes for Homework 4, due Feb. 19

Notes for changing degrees to radians and vice versa



Notes for trig values for angles in Quadrants II, III and IV
 

Notes for Homework 3, due Feb. 12

Notes for finding all five other trig values at an angle, give sine or cosine or tangent

Notes on rationalizing denominators
 

Notes for Homework 2, due Mon. Feb 4

Notes for defining a triangle by three points in the plane. Finding the area and classifying the triangle by finding the distance of all three sides.

Dealing with "the famous angles": 0°, 30°, 45°, 60°, 90°. Finding secant, cosecant and cotangent of these angles.
 

Notes for homework #1, due Jan. 29

Notes on the triangle inequality

Classifying triangles defined by side lengths
Let the three sides be called a, b and c, where c is the longest side.

There are three possible relationships between the sum of the squares of a and b and the value c². 

a² + b² > c²   This means the triangle is acute.
a² + b² = c²   This means the triangle is right. This is the Pythagorean Theorem,
a² + b² < c²   This means the triangle is obtuse

Here are some practice problems
     

Given one angle measurement, create an isosceles triangle. If the angle is acute, there are two possible isosceles triangle.

Notes on Heron's Formula and practice problems