PreCalculus Generic Notes
© by Scott Surgent
Law of Sines
An oblique triangle is one without a right angle. While you may have perceived
trigonometry to require a right triangle, the law of sines and the law of cosines allow us to
solve for any remaining unknown angles or sides, for any triangle, as long as we are
given some basic required information.
First of all, each triangle will have sides a, b and c, and angles A, B and C, such that side
a is opposite angle A, and so forth.
Next, we need to know a certain amount of information. For the law of sines, the
following two conditions are (independently) necessary requirements:
1.
2.
We need to know the measure of two angles and one side (abbreviated SAA, ASA
or AAS).
We need to know the measure of two sides and an angle opposite one of them (not
the angle between them). This is SSA.
If one of these two conditions is met, then we can use the law of sines to set up easy
ratios to solve for the remaining sides and angles.
The law of sines. states that
used as well.
a
sin A
=
b
sin B
=
c
sin C
. Equivalently, the reciprocal form can be
The proof is simple. Most texts use a generic drawing, you’ll note that sin B = ha , so that
h = a sin B .
Also, sin A = bh , so that h = b sin A .
Therefore by transitivity,
a sin B = b sin A . Divide the equations by sin A and sin B, we get sina A = sinb B . A similar
argument can be shown for
c
sin C
by dropping a perpendicular from A or B to create sin C.
Please look an example of AAS. Remember: Since you know two angles, you can easily
get the third since the sum of the angles equals 180 degrees. (Many of these problems are
done in degree mode).
The case SSA can possibly be ambiguous: 0, 1, or 2 solutions are possible. Please see the
examples for the conditions under which each occur.
Law of Cosines
The law of cosines allows us to solve situations when we know the following:
1.
2.
The measure of all three sides (SSS), or
Two sides and the angle between them (SAS)
The law of sines does not work here since you can’t form a complete ratio, leaving too
many variables to solve for.
PreCalculus Generic Notes
© by Scott Surgent
The proof of the law of cosines is probably given in the text’s appendix. I suggest you
look it over closely since each step involves something you have done before. There are
three standard forms of the law of cosines, depending on the unknown. Notice that if you
cover up the last term (for instance the “-2ab cos C”), you have the Pythagorean theorem.
The law of cosines is often called the generalized Pythagorean theorem.
Study each of these examples in this section. All are good and illustrate how to approach
this concept. Ultimately it’s just number crunching..
Heron’s formula is a clever formula (from antiquity) that allows you to find the area of a
triangle given the length of the three sides. I won’t insist that you memorize it.
Vectors in the Plane
The next natural step after you have a solid grounding in trigonometry involves vectors.
A vector is a directed line segment with a direction and a length (magnitude). The
placement of the vector in a plane is irrelevant. If two vectors have the same magnitude
and direction, they are considered equivalent.
⎯⎯→
The usual notation for a vector is to write PQ , where P is the initial point and Q is the the
terminal point. The length of the vector is denoted PQ . Also, note that boldfaced
lower case letters are used to denote vectors, such as u and v.
A convenient method to denote a vector numerically is to write u = a, b , which means that
from the initial point, you are to go horizontal a units, then vertical b units.
To add two vectors, simply sum the horizontal units and the vertical units. Subtraction
works in a similar way. A scalar multiple of a vector is expressed by the form ku, where
k is any number. The term scalar is used to denote a number, in order to differentiate it
from a vector. For example, if u = 3,−2 (a vector), then 2u = 6,−4 (the 2 is a scalar;
very intuitive.) You will note that the new vector is twice as long as the original vector.
A unit vector is a vector with magnitude 1. Given any vector u, we can form its unit
vector by dividing by its magnitude, hence u/||u|| now has length 1.
Since vectors have direction, it is helpful to use angles to describe their direction relative
to a horizontal or another vector. The arctan function is useful for determining the angle
θ.
A vector can be described by its components. This process requires us to decompose the
vector into its horizontal displacement and vertical displacement. For instance, if a
vector has length 30 feet and a direction angle of 20 degrees, how far horizontally must
PreCalculus Generic Notes
© by Scott Surgent
one go, then how far vertically must one go, to locate the terminal point? This concept is
particularly helpful when talking about bearings on terrain, or force vectors.
Vectors and Dot Products
The dot product of two vectors is to multiply the corresponding entries then add the
products. It is important to remember that the dot product of two vectors is a scalar.
The dot product is actually proven using the Law of Cosines. The dot product allows us
an easy way to quickly calculate the angle θ between the two vectors. Because the cos θ
is positive if 0 ≤ θ ≤ π2 and negative if π2 ≤ θ ≤ π , the sign of the dot product offers a
quick way to tell if the two vectors are acute (the dot is positive) or obtuse (the dot is
negative). If the dot product is zero, the two vectors are perpendicular and we say the
vectors are orthogonal.
In physics, forces are often decomposed into their components, which allow one to
calculate the forces by viewing them separately as “up-down” and “left-right” forces.
Hint: it often makes most sense to decompose a vector into its natural “x” and “y”
components, if possible.