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This is also one more important mathematical formula in multiple and sub-multiple angles’ trigonometry. We have previously developed summation of sine, summation of cosine and differentiation of sine function with two different angles. Similarly, we are going to derive differentiation of Cosine function.

First of all we have to assume, C and D are different angles of a right angle triangle. Now we are going to subtract these two angles with respect Cosine, which means we are developing simplification of below trigonometric function.

CosC – CosD

It is very complicated or not easy to simplify this trigonometric function directly by using mathematical applications. However, we can simplify this function by taking an assumption. Consider C and D are equal to A + B and A – B respectively for our convenience.

Developing a mathematical fundamental of Cosine with two different angles into its sub multiple angles help us to solve some complicated problems in trigonometry. Actually, these types of functions are basic steps to develop advanced trigonometric formulas. In other words, this trigonometric formula helped us in developing more advanced formulas, saving time and solving difficult problems easily in few steps.

We have also derived similar mathematical fundamental but it is in Sine function. Now, we are going to learn the basic procedure to express this formula into its sub multiple angles. The basic Cosine function’s two imaginary angles are C and D of a right angle triangle and the respective function would be like this.

CosC + CosD

We have learned previously the summation of two Sine functions with two different angles. This concept is called as multiple and sub multiple angles’ trigonometry. In this article, we are going to express the subtraction of two Sine functions with two different angles. In order to prove this, we have to assume C and D are two different angles of a right angle triangle. And the subtraction is as stated below mathematically.

SinC – SinD

We can convert the above trigonometric equaion with mathematical fundamentals. However, there is an alternative method to simplify this funtion easily. We have to consider one assumption in order to covert multiple angles into its sub multiple angles.

C = A + B and D = A – B

We have learned previously, trigonometric functions are depends, which means trigonometric functions such as Sine, Cosine, Tangent, Cotangent, Secant and Cosecant always depend on a variable and the variable is called as an angle. Previously, we have developed several mathematical formulas with single angle.

However, we are going to learn, trigonometric functions with different angles which sometimes called as multiple angles and sub multiple angles. These multiple and sub multiple angles’ trigonometric relationships well help us in dealing very complicated problems easily, developing more trigonometric formulas and improving knowledge on problem solving.

Most of the people do not have much knowledge on multiple and sub multiple angles functions. Now, I would like to explain multiple and sub multiple functions clearly.

This is most useful mathematical formula in trigonometry. Actually, this formula tells relationship between two multiple angles. In this trigonometric function C and D are two multiple angles of a right angle triangle. Now, our main objective is two simplify summation of two similar trigonometric functions with two different multiple angles, which means we have to simplify below trigonometric function mathematically.

SinC + SinD

It is not easy to find the solution for this trigonometric equation directly, which means we have to approach in different way in order to find the required solution.

Let us assume C = A + B and D = A – B

In the assumption, we are considering that the summation and subtraction of two different angles give multiple angles C and D. Now submit C and D value in terms of A and B angles.

Logarithms is a most useful mathematical concept in mathematics. Now we have scientific calculators and very powerful computers. So, we can easily solve any numerical problem easily within few seconds. We are still developing our technology to save time.

However, the scenario was entirely different initially. Scientists face lots of problems with complicated problems. That’s why we are separately learning mathematics. Mathematicians put pull stop to those problems by introducing new mathematical concepts. Logarithms is one those mathematical concepts.

Actually, logarithms solve very complicated numerical problems which contain big numbers and decimals. Similarly, logarithms are useful in exponential series. Now you understood why we are giving priority to logarithms. Later, you will know everything completely. In this mean time, let us discuss most useful basics of logarithms.

Pythagorean Numbers are most commonly used numbers in trigonometry. Actually, these numbers are determined with the help of right angle triangle. In other words these numbers are actually determined with Pythagoras theorem.

According to the Pythagoras theorem

(Hypotenuse)² = (Opposite)² + (Adjacent)²

Take numerical numbers 3 and 4 as opposite and adjacent sides’ values. With the help of Pythagoras theorem we can find the hypotenuse value of the respective right angle triangle.

Before going to talk about article writing, we have to know meaning and purpose of an article. Article means information which educates the people by providing quality content on any topic. The main purpose of an article is to help people to learn something from our content.

Everyone can become article writer by exposing knowledge and expressing inner feelings in words. However, everyone’s information may not be called as article. For example, we have been listening several songs but only few songs touch our heart. Similarly, we have read several informative articles but only few authors can satisfy us with best information.

Should we have some qualities to become best article writer?

My answer is yes, person who wants to become best article writer should have four qualities. Then only the respective person can show his/ her specialty to readers. These are the list of four essential qualities we should have, to write best and popular article.

1. Interest
2. Complete concentration
3. Thinking in readers’ point of view
4. Writing style

This is most useful trigonometric formula. We have already derived similar mathematical fundamental but it is in Sine terms. In the same way, we can also derive Cosine function with multiple angles. A, B and C are multiple angles of a right angle triangle.

Deriving the entire mathematical function is little bit complicated. However, we can reduce the difficulty by assuming two angles as one angle, which means let us assume an imaginary angle θ. It is nothing but summation or resultant angle of two angles. Θ = A + B. As per this assumption, we can write our Cosine function as stated below.

Cos(A + B + C) = Cos(θ + C)

We have already derived a two angle Cosine formula. Now, we have to apply that formula in order to further simply of this function.

Cos(A + B) = CosA.CosB – SinA.SinB

This trigonometric formula may not useful in entire trigonometry. Sometimes, this type of mathematical expression could be useful in trigonometry problem solving. So, we are going to derive this advanced trigonometric formula. Actually, we have successfully derived two angles formula but here three angles A, B and C are come into picture.

However, in the same way we can also prove this theorem. In order to simpler this function we have to consider two angles as one angle, which means consider A + B as one angle and the resultant angle is θ. In other words, here θ is equal to A + B.

Sin(A + B + C) = Sin(θ + C)

We have already proved one trigonometric formula previously. Now, we are going to apply that mathematical application here in order to simplify this function.

Sin(A + B) = SinA.CosB + CosA.SinB