Notes for Homework 5

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

Notes for Homework 4

Notes on finding sine if you are given cosine. Once you find those, the other four trig function can be defined in terms of the first two. Also, notes on finding sine and cosine if you are given tangent.

Notes on changing from degrees to radians and vice versa.

Notes for Homework #3

Notes for getting rid of radicals in denominators of fractions

Notes for finding all six trig functions for an angle if you are given sine, cosine or tangent, assuming we are in Quadrant I

Notes for Homework #2

Notes for defining triangles by angles in degrees 

Notes for defining triangles by three points in the plane, finding the area and the classifications by finding the distances
 

Notes for Homework #1

Notes for defining triangles by side lengths.

Notes for degenerate triangles.

Notes for Heron's formula.
 

Degenerate triangles

The triangle inequality state that if the sides of a triangle are labeled a, b and c, we need two inequalities to be true a + b >= c >= |a - b|. If we get an equality, the triangle is said to be degenerate. For example, if the lengths are 9, 6 and 3, we would draw a line segment of length 9 with a point on the line splitting into two parts of length 6 and 3. To think of it in terms of base and height, the base would be 9 and the height 0, so the area would be zero.

There are some facts about degenerate triangles you should know.

a) If you plug in these values into Heron's formula, you will get an area of 0.

b) The angle sum is 180°, since there are angles of 180°, 0° and 0°. It is not truly obtuse, since 180° is a straight angle.

c) If the two short sides are equal, it qualifies as an isosceles triangle, if they are not equal, it is scalene.

d) When we define a triangle by three distinct points in the plane, we will get a degenerate triangle if the three points are all on the same line, which is known as colinear.

 

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