It is not the intention of this researcher to include the sum-total of all known trigonometry in one Guide entry. There are, however, a few fundamental concepts in this area of study that are worthy of note simply for the facts that they are mathematically elegant results and that their practical uses in a myriad of different fields goes without further mention. It would not seem worthy to simply state these concepts without some attempt to argue their validity, so an elementary knowledge of the basics of right-angle trigonometry is required in order to appreciate the meaning of the following justifications. The formulae generated by these concepts, however, require little more than a pocket calculator.
In utilising the Laws of Sines and Cosines, it is usually assumed that problems are presented wherein at least 3 of the 6 quantities (6 being the 3 sides together with the 3 angles) for the various triangle measures are given, or at least implicitly obtainable from the given information. This is hardly a detraction from their real-world uses, in which triangles exist across rivers, through outer space, or wherever people choose to connect the dots that they can't always reach from a physical standpoint. In these cases, surveyors and/or scientists are able to measure some, but not all of the quantities they need. They are usually able to measure just enough to let trigonometry do its thing...and with that, what will follow are a few samples of the things it can do.
The Law of Sines can be used to completely determine the measures of the sides and the angles in a given triangle, provided the given triangle information is: A) the measures of any 2 angles and a length of any one side, or B) the measure of any one angle and the side opposite it together with one other side. However, in scenario B exists the possibility of 0, 1, or even 2 triangles satisfying the given information. This is the aptly-named "ambiguous case" in trigonometry.
Scenarios in which the Law of Sines is not a powerful-enough tool for completely determinining a triangle's measurements occur if the given information about the triangle is: A) the measures of the 3 angles, B) the lengths of the 3 sides, or C) the lengths of any 2 sides and the angle they form. In scenario A, there are an infinite number of solutions based upon the geometric concept of similar triangles, which occur when corresponding angles of differently sized triangles are congruent. For cases B and C, it is the Law of Cosines which must first be utilised in order to determine enough aditional information about the triangle in order to complete the problem using the easier Law of Sines.
The process of labelling the vertices, sides, and angles of triangles with various letters for notational and referral purposes in trigonometry is kept consistent from triangle to triangle. Since the Laws of Sines and Cosines are written with this notation in mind, it is best to introduce the various letters before the Laws.
Let triangle ABC be any triangle. In more mathematically explicit language, for the purposes intended here: A, B, and C are 3 given points in a 2-dimensional space, not all colinear, such that each point is at a vertex of the triangle formed by pairwise connection of the given points with line segments.
Let the lowercase Greek letters α, β, and γ refer to the angles within triangle ABC with vertices at the corresponding uppercase english letters.
Let lowercase letters a, b, and c refer to the lengths of the sides of triangle ABC, where side a is opposite angle α, side b is opposite angle β, and side c is opposite angle γ.
The Law of Sines
One of the two fundamental relationships which holds true for any given triangle is the Law of Sines. This law asserts that:
a/sin α = b/sin β = c/sin γ , or equivalently, (sin α)/a = (sin β)/b = (sin γ)/c.
Proof - To facilitate the ease of proof, let triangle ABC be situated in an xy-coordinate place (also called a Cartesian coordinate plane in honor of French mathematician Rene Descartes) in such a way that vertex A coincides with the origin and vertex B lies on the positive x-axis. In this position, α is now an angle in "standard position" (a trigonometric term for an angle with one ray of the angle coinciding with the positive x-axis in an xy-coordinate plane), with vertex C either lying above or below the x-axis.
Without loss of generality, it may be assumed that C lies above the x-axis (for if it did not, the reflection C' of point C across the x-axis could be utilised, and the validity of the proof for triangle ABC' would extend to the "mirror-image" congruent triangle ABC). Also note that segment AC is equivalent to side b in triangle ABC, and segment BC is equivalent to side a.
Letting D be the point on the x-axis that is also on the same vertical line as vertex C, both triangles ACD and BCD are right triangles (triangles containing 90° angles) with right angles each at vertex D.
From concepts in right-angle trigonometry (namely, the fact that the sine of an acute angle in a right triangle is the ratio of the side opposite to the hypotenuse), in triangle BCD: sin β = CD/BC = CD/a, hence CD = a*sin β.
In triangle ACD, angle α is either interior or exterior to the triangle. If α is interior to the triangle, then: sin α = CD/AC = CD/b, hence CD = b*sin α. If α is exterior to the triangle, then the angle interior to the triangle at vertex A has a measure of 180°-α, and by the subtraction formula for sine, sin (180°-α) = (sin 180° * cos α) - (cos 180° * sin α). But sin 180° = 0 and cos 180° = -1, thus sin (180°-α) = sin α, and the result is again obtained.
Substituing for CD in the results obtained from both triangles, the equality a*sin β = b*sin α is obtained. Dividing both sides by a*b yields the desired result, which can then be generalized to the three-part Law of Sines based upon the ability to re-orient the given triangle into the necessary position. Q.E.D. (Quod Erat Demonstratum - Latin for "which was what was to be demonstrated")
The Law of Cosines
The Law of Cosines is the second of two fundamental relationships which holds for all arbitrary triangles. It is more complex than the Law of Sines, however one saving grace in its favor is that once the Law of Cosines is utilised once to determine one additional piece of information about a triangle, then enough information will be known about the triangle at that point to be able to use the more simplistic Law of Sines to determine the remainer of the triangle.
Using the same notation as before, the Law of Cosines asserts that:
a² = b² + c² - 2bc*cos α.
This may also be stated in the following forms:
b² = a² + c² - 2ac*cos β, and c² = a² + b² - 2ab*cos γ.
Three forms are given, even though one would suffice. This is done to bring forward the fact that the notation itself is irrelevant, it is the relationship of the 4 variables in the equation that remains consistent. In regular language, the Law of Cosines states that The square of any one side equals the sum of the squares of the other two, less two times their product times the cosine of the angle opposite the first side. Reviewing the pattern of each of the 3 versions of the Law will make this clear, and it is quite often easier to commit this lengthier Law to memory by memorising the relationship as opposed to the 3 distinct formulae.
Proof - The first version of the Law of Cosines will be proven, and without loss of generality, that result may then be extended to the other two variations. Let triangle ABC be situated in an xy-coordinate plane in the same manner that was established in the first 2 paragraphs of the proof for the Law of Sines. Again, let D be the point on the x-axis and on the same vertical line as C. Let the coordinates of C be represented by the xy-coordinate pair (d,e).
There are specific theorems established in elementary trigonometry which prove the equality of trigonometric function values to ratios determined by point coordinates divided by distances from said points to the origin. Specifically, if an arbitrary point P(x,y) lies in a coordinate plane and is a distance r from the origin, then the ray from the origin through P together with the positive x-axis determine an angle θ in standard position, in which sin θ = y/r and cos θ = x/r. In the Law of Cosines diagram, point A coincides with the origin, and segment AC (side B) together with segment AB (side c) on the positive x-axis determine an angle (in this case, angle α) in standard position. Using the chosen variables for the coordinates of point C,
cos α = d/b , and sin α = e/b,
from which we find the following values for coordinates d and e:
(d,e) = (b*cos α,b*sin α).
Since segment AB on the positive x-axis corresponds to side c and A lies at the origin, then B can be said to have xy-coordinates (c,0). Segment BC corresponds to side a, and can be determined by using the distance formula on points B and C:
a² = d(B,C)² = (d-c)² + (e-0)²
= (b*cos α - c)² + (b*sin α)²
= b²cos²α - 2bc*cos α + c² + b²sin²α
= b²(cos²α + sin²α) - 2bc*cos α
= b² + c² - 2bc*cos α. Q.E.D.
It is sometimes more useful to have the Law of Cosines in a form in which the solution for one of the angles is in terms of the 3 sides. As an example of how the Law of Cosines can be manipulated, if one solves for the angle α in the first given version of the Law of Cosines, the solution is
α = arccos[(b² + c² - a²)/2bc],
where "arccos" refers to the inverse cosine function. In this form, the values of the 3 sides are easily plugged and the value of α is quickly determined with the use of a standard calculator.
HOWEVER : There is a caveat in using the Law of Cosines. Namely, if the given pieces of information for a triangle are lengths of the 3 sides, it is imperative that the first angle found be the angle opposite the longest side first. The reason for this is that there is always the possibility that one of the angles in the triangle will be obtuse (meaning that its measure is more than 90 degrees). If there is going to be any obtuse angle in any triangle (at least in Euclidean geometry), then it will be opposite the longest side. If one of the smaller angles is found first, and the Law of Sines is used to finish the problem, there exists the possibility of obtaining incorrect solutions if one of the angles of the triangle turns out to be obtuse.