Solving Trigonometric Equations Solving Trigonometric Equations tan2 t + 1 = sec2 t equation (1) is a trigonometric equation. Here tan, sec is trigonometric functions. We know in trigonometry there are six trigonometric functions, Sin, cos,tan cosec ,sec , cot. A trigonometry equation that hold true for any angle which is called trigonometric identity, and these angles are true for certain angles which is known as conditional Trigonometric Equations. Now we will see some techniques for solving trigonometric equations and also we will learn how to derive the equations an infinite number of solutions to an equation based on a single solution to that trigonometric equation. Trigonometric equations can be solved easily without using a calculator. Like if we Want find value of sin 300 then we can solve the value of sin 300 =1/2 =.5 . but some condition like tan (x)=3.2 such equations has no simple answer that will have to memorized. It would we tedious to use calculator to finding this value.
Know More About :- Derivative Of Secx
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For above problem like these, the inverse trigonometric functions are helpful. The inverse trigonometric functions are also same as trigonometric functions except x and y value which is reversed form. For example cos(y)= x is y = arccos(x). here arccos is not a function because this is assign more than one element and that element have a range to each element of the domain of function. For example if we find the all solutions of sin(y)=1/2 then solution is y=30 degree, 150 degree, 390 degree, and so on. If we solve this value by inverse trigonometric functions, normally it is very difficult, but it becomes possible by calculator. Now we are going to discuss the solution of trigonometric equations, for solving a trigonometric equations first we use to trigonometric quantities and algebraic techniques to reduce the given trigonometric equation to an equivalent expression. So that was the proper explanation of trigonometric equations now we are going to learn how to solve trigonometric equation. Example:1 Find the all possible solutions values of given trigonometric equations cos x + √2 = -cos x where interval [0,2∏] Solution: For finding all solutions we need to follow the below steps: Step 1: In the first step we write the given trigonometric equation. Cos x +√2 = -cos x
Learn More :- De Moivre s Theorem
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Step 2: In this step we solve the trigonometric equation for finding value of cos x cos x + √2 = -cos x cos x + cos x + √2= cos x + cos x = -√2 2 cos x = -√2 cos x = -√2/2 after simplification we got the value of cos x. Step 3: Now we solve the value of cos x for finding the value of x . cos x =-√2/2 we know cos x has period of 2∏ Step 4: In this step we we calculate all solutions in given interval In this problem we have given interval [0,2∏] so we find the value between 0 to 2∏ Now we got the value x = 3∏/4 and x =5∏/4 . In conclusion we add multiple of 2∏ to each of getting values Now the general form of x = 3∏/4 +2n∏ and x = 5∏/4 +2n∏ Here n is an integer. So that was the article Trigonometric equation. I hope you will understand this article and in the next article we will learn other properties of equation.
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