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.
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.
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
We have learned reciprocal property of tangent and secant functions and they are indirectly related each other. Now we are going to learn the reciprocal property of Cotangent and Cosecant and the relationship is almost similar to that. In other words, these two functions are also related indirectly just like those functions.
The reciprocal property of Cotangent and Co-secant can be learned with the help of trigonometric identities, which means we have already proved that
Cosec2θ – Cot2θ = 1
This trigonometric identity can be split into two different trigonometric functions for our convenience.
Trigonometric functions Secant and Tangent have reciprocal relationship but it is not direct relationship, which means tangent and secant functions are indirectly related. We are going to learn how they are related indirectly in this post.
In previous post, we have learned trigonometric identities. In other words, we have successfully proved that
Sec2θ – Tan2θ = 1
This trigonometric identity could be split into two different mathematical functions as stated below.
(Secθ – Tanθ).(Secθ + Tanθ) = 1
