How to solve a School Math problem in a few simple steps, Trigonometry 3

A few simple steps to solution compared to conventional complex procedure

How to solve a school math problem in a few simple steps trigonometry3

Many times we find at high school level, math problems are solved following a long series of steps. This is what we call conventional approach to solving problems.

This approach not only involves usually a large number of steps, in most cases, the steps themselves introduce a higher level of complexity.

These two aspects of conventional problem solving together result in what we call costly and inefficient problem solving.

In school math or real life problems, this is the path commonly followed by most. In school math, among other topics we find this form of solution abundantly in case of problems of the type,

Prove that, "Some expression " = "Some other expression".

In school math terminology, the "Some expression" is called LHS (short form of Left Hand Side) and the "Some other expression" as RHS (short form of Right Hand Side). This type of problems occurs aplenty in Elementary Trigonometry of proving Identities.

We have covered two such examples already in,

How to solve a School math problem in a few simple steps, Trigonometry 1, and

How to solve a School math problem in a few  simple steps, Trigonometry 2.

This is the third example.

In each of these examples we will go through the reasoning process of reaching the solution in a few simple steps using inherent problem solving abilities of the student and taking as little help from routine Maths constructs as possible. Usually such solutions turn out to be most effficient with least memory load.

In various types of these problems we will show how to reason in diverse situations always with the target of choosing the shortest and most elegant path to the solution.

In contrast the long and mostly complex solutions use a conventional approach of going towards the solution from LHS to RHS (or initial state to goal state) through many steps using the expansion of the LHS expression and then simplification or consolidation of the numerous expanded terms towards the form of expression on the right hand side, that is, the RHS.

This approach has two important disadvantages,

  1. Not only does this approach take considerable amount of time and effort, but because of large number of steps and increased complexity, chances of error is much higher in this approach.
  2. This mechanical approach relies heavily on manipulation of terms using low level mathematical constructs without using the problem solving abilities of the student. In fact, if students follow only this approach for solving problems, they may tend to become used to mechanical and procedural thinking suppressing their inherent creative and innovative out-of-the-box thinking abilities.

Let us go through the process of solving another Trigonometric Identity problem to appreciate the difference between the conventional approach and the problem solver's intelligent approach. The difference is always significant and measurable.

Problem example

Prove the identity:

$\displaystyle\frac{sin\theta}{1 + cos\theta} + \displaystyle\frac{1 + cos\theta}{sin\theta} = 2cosec\theta$

First try to solve this problem yourself and then only go ahead. You might be able to reach the elegant solution to this problem yourself.

Conventional solution

Usually a conventional approach in this case will proceed to combine the two fractional terms using deductive process involving quite a bit of mathematical manipulation.

Flllowing this common approach, we combine the two terms in a brute-force method (there can be other such conventional methods) to transform the LHS as,

$\displaystyle\frac{sin\theta}{1 + cos\theta} + \displaystyle\frac{1 + cos\theta}{sin\theta}$

$\displaystyle\frac{sin^2\theta+ (1 + cos\theta)^2}{sin\theta(1 + cos\theta)}$

$ = \displaystyle\frac{sin^2\theta + 1 + 2cos\theta + cos^2\theta}{sin\theta(1 + cos\theta)}$

$ = \displaystyle\frac{sin^2\theta + cos^2\theta + 1 + 2cos\theta}{sin\theta(1 + cos\theta)}$

$=\displaystyle\frac{2 + 2cos\theta}{sin\theta(1 + cos\theta)}$

$=\displaystyle\frac{2}{sin\theta}$

$=2cose\theta$

In this method we went through a straightforward approach of combining the two fraction terms in the LHS, and going though carefully the simplification process relying on our knowledge of conversion of trigonometric terms so that we can move towards the end state of the given expression with certainty and no error.

As such there is nothing wrong with this conventional straightforward approach.

On second thoughts though, we find we have to go through a quite a bit of routine trigonometric procedures with no use of our innovative skills in finding new approaches.

In the discipline of Efficient Math Problem Solving we emphasize repeatedly, that

Always use your efficient problem solving strategies more and the routine mathematical procedures less in solving any math problem.

This is broadly our Less facts and more procedures approach that we would deal with in detail later. Nevertheless, each of the efficient math problem solving examples actually use this more abstract problem solving approach. Formally we state this Less facts and more procedures approach as,

For solving any problem, use least amount of unconnected facts and more of interlinked concepts and efficient problem solving procedures.

In maths problem solving, this approach boils down to the strong recommendation highlighted by many eminent mathematicians among others,

Do more problem solving and less maths.

Efficient solution in a few steps

Instead of this conventional approach, the very first step that you must take as usual is to analyze the problem.

In any problem solving, math or otherwise, this must be the first step. You must start with analyzing the problem statement.

Without the first step of Problem analysis, no efficient problem solving is possible.

A corollary,

In competitive exams, and also in competitive work environment, the first step of problem analysis is crucial for success.

The better and quicker you are able to analyze a problem, the faster you would reach the desired solution.

Problem analysis

As a first step of analysis in such math problems, we follow the End State analysis approach invariably and compare the desired end state expression with the given initial state expression for detecting similarities or dissimilarities.


Aside: Psychology and process of problem solving by End State Analysis: The desired goal to reach undoubtedly rank highest in importance in your mind among all other information about the problem as your natural tendency is to reach the goal state in quickest possible time.

This pre-eminence of importance of the desired end state or goal state focuses your attention naturally on this end state when you know the desired end state. This is the case of proving identities. 

What would you look for in the end state?

If it is a journey from one city to another, you study the distance from your starting point to the destination. You try to judge what kind of transportation along which possible path would take you to the destination in shortest possible time, isn't it? We assume here the importance of optimal journey, which is the case of any important problem solving.

The same happens in this case. You judge the end state (or RHS expression) with respect to the initial given state (or LHS expression). If somehow you find significant similarities between the two, it would be easy for you to span the gap between the two states quickly.

In all cases though there would be significant dissimilarity between the initial starting point and the desired end point. The similarity would invariably be there but it would be hidden from casual inspection.

This is where the ability of key information discovery plays its prime role in solving the problem.

More often than not, ability to recognize useful common pattern, even if hidden, results in key information discovery.

If you don't know the desired goal state, from initial problem analysis you have to form possible desired goal states.

We add one aspect here, while studying the similarity between the end state and the intial state, many times the significant dissimilarity also may help you to find the key to the problem solution.


In this problem we observe, though the terms of the given expression are fractions involving denominators, the end state goal expression on the RHS does not have any fractions. This is the significant dissimilarity that would guide us to our main strategy to deal with problem elegantly.

Key information discovery

Thus we form our goal as transforming each of the two terms in the LHS to eliminate the denominators as quickly and as elegantly as possible, so that there is no fractional terms in the simplified expression. It must be so as the RHS also does not have any fractional term with denominators.

This objective of eliminating denominators is one of the most frequently encountered goal in simplifying various expressions in various branches of mathematics.

In all such simplification processes a set of helpful constructs or techniques provide immense help. Some of these resources are specific to the branch of mathematics, but some are general and can be applied in any branch of mathematics. Let's briefly go through what help we have in Trigonometry for eliminating denominators.

Helpful constructs in Trigonometry for eliminating denominators

A. Forming denominaor as a factor in the numerator

Often ignored but one of the easiest methods of eliminating denominator is to form the denominator as a factor in the numerator either in product form or in a sum form. In both cases we would achieve good amount of simplification.

Example:

Simplify $\displaystyle\frac{1 + cos\theta - sin^2\theta}{1 + cos\theta}$.

Solution:

$\displaystyle\frac{1 + cos\theta - sin^2\theta}{1 + cos\theta}$

$=\displaystyle\frac{cos\theta + cos^2\theta}{1 + cos\theta}$

$=cos\theta$.

Here we have formed the factor of denominator in the numerator by transforming it using the most important Trigonometric construct,

$sin^2\theta + cos^2\theta = 1$, Or, $1-sin^2\theta = cos^2\theta$.

These constructs are very frequently used in simplifying Trigonometric expressions.

B. Rationalizing the denominator

While rationalizing a denominator we multiply the numerator and denominator both by a suitable factor so that the denominator gets eliminated.

In simple rationalization, we invariably use another highly useful algebraic construct,

$(a + b)(a - b) = a^2 - b^2$, and if $a^2 - b^2$ is a very simplified term we achieve elimination of denominator by this process.

Let us take the example of the denominator of our problem iteslf.

Taking then the first term itself we have,

$\displaystyle\frac{sin\theta}{1 + cos\theta} $

$= \displaystyle\frac{sin\theta(1 - cos\theta)}{1 - cos^2\theta}$

$=\displaystyle\frac{sin\theta(1-cos\theta)}{sin^2\theta}$, observe how $sin^2\theta + cos^2\theta = 1$ comes of help,

$=\displaystyle\frac{1-cos\theta}{sin\theta}$

Now we stop and have a look at the second term and the desired expression. This is End State Analysis in action.

After going through a well analyzed and intended step, we stop and analyze again before taking the next step.

If our initial analysis were in the right direction, we would be able to see the solution up ahead only a few steps later that would take a few tens of seconds.


Our simplified expression is then,

$\displaystyle\frac{sin\theta}{1 + cos\theta} + \displaystyle\frac{1 + cos\theta}{sin\theta}$

$= \displaystyle\frac{sin\theta(1 - cos\theta)}{1 - cos^2\theta} + \displaystyle\frac{1 + cos\theta}{sin\theta}$

$= \displaystyle\frac{sin\theta(1 - cos\theta)}{sin^2\theta} + \displaystyle\frac{1 + cos\theta}{sin\theta}$

$= \displaystyle\frac{1 - cos\theta}{sin\theta} + \displaystyle\frac{1 + cos\theta}{sin\theta}$

$= \displaystyle\frac{2}{sin\theta} $

$=2cosec\theta$.

Observe how one simple reasoned act of eliminating or simplifying the denominator gives you the solution in practically one step.

Note: Here we say "eliminating or simplifying the denominator". The reason is, if you can transform the denominator in a single term expression, in most cases it is as good as eliminating the denominator itself. The denominator creates problems as long as it is a sum of terms.

Deductive reasoning sum up

On comparison of the RHS and LHS we find the given expression having  the first term with sum of terms as a denominator while the goal state expression is without any fraction term involving denominators.

As the second term denominator in fact is the RHS term $cosec\theta$ itself, we don't bother with it at this moment. We decide the best approach is to  eliminate the denominator of the first term by rationalization.

After rationalization and minor simplification when we compare the two terms again, we find the second terms of the two terms cancelling out leaving $2cosec\theta$ as in the RHS. One single act of rationalization gives us the solution.

End note: This is not the only example of this type. You would find very large number of this and other type of problems where you would invariably be able to improve upon the time and effort in conventional solution.

Our recommendation: Always think: is there any other shorter better way to the solution?