Technology
Introduction to Trigonometric Ratios | Episode 10
In this episode of Mathematics Simplified, host Anjali Sharma explores the domain and range of trigonometric functions, breaking down complex concepts into easily digestible parts. Listeners will gain...
Introduction to Trigonometric Ratios | Episode 10
Technology •
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Interactive Transcript
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Hello there friends! Welcome to another episode of Mathematics Simplified. The podcast
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where we make math easy and fun. I'm your host Anjali Sharma and today we are diving
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into a very important topic in trigonometric functions that is domain and range of trigonometric
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functions. This is a topic that confuses a lot of students but by the end of this episode,
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I'm sure you will have complete clarity. First thing first, what exactly is domain and range?
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Let's start with the basics. Domain of a function means all the possible values that the input
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can take up and for which the function is well defined. Range means all the possible
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output values the function can take. For example, if I say fx is x square, the domain is
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all the real numbers because I can put any number in place of x that is any real number.
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So x square is always well defined. The range is all known negative real numbers because x
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square is always greater than or equal to 0. So what are the domain and ranges for the trigonometric
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functions? Let's take a look at each one of the trigonometric functions one by one.
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The very first one is the sine function that is y is equal to sine x.
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Sine function is defined for all the real values of x. So the domain is all the real numbers
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and in case you want to show it as an interval, it goes from negative infinity to positive infinity.
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And the sine function can only take values between negative 1 and positive 1.
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Inclusive of both these values, so the range is closed interval minus 1 to 1.
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Similarly, for the cosine function, the range and the domain are defined exactly the same.
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That is, the domain is all the real numbers and the value of the function lies between negative 1
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and positive 1. That is ranges, closed interval negative 1 to 1.
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For the tangent function, that is y is equal to tan x, this one is a bit tricky.
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Remember that we can write tan of x as sine x divided by cos of x.
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So wherever cos of x is 0, tan x is not defined at all.
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Cos x is 0 for pi by 2, 3 pi by 2, 5 pi by 2 or in fact any odd multiple of pi by 2.
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So what happens to the domain of tan x? It is all the real numbers except odd multiples of pi by 2
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and range of the function is all the real numbers. That is, the open interval negative infinity
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to positive infinity.
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What happens to the code tangent function? It is also defined as cos of x divided by sine of x.
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So it is not defined when sine of x is 0. So that happens when x is either 0 for pi or 2 pi
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or in general it is n pi, wherever n is an integer.
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So the domain of this function that is the code function or the code tangent function is
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all the real numbers except n pi. Here n is an integer.
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The range of the function is all the real numbers.
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For the secant function that is the one upon cos function that is not defined wherever cos is 0.
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So once again the domain is all the real numbers except odd multiples of pi by 2.
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The range is a bit special because since the secant function is either greater than or equal to 1
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or it is less than or equal to negative 1. So the range is union of the sets. Negative infinity
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to negative 1 union of that with the interval 1 to infinity.
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For the cos secant function whenever sine x is 0 the function is thought to find.
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So what happens to the domain? It is all the real numbers except x as n pi.
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That is all the multiples of pi and the range is once again negative infinity to negative 1.
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Here negative 1 is included and union of that with 1 to infinity. Once again one is included here.
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So you can make a summary of all of that so that you can quickly revise it at any point of time.
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The functions are sine x. Domain is all the real numbers, ranges negative 1 to 1. Both numbers
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inclusive. Cos axis once again domain all the real numbers, ranges negative 1 to 1 both the end
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points included. The next domain is all the real numbers except odd multiples of pi by 2
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and ranges all the real numbers. For cot function domain is all the real numbers except
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multiples of pi, ranges all the real numbers secant x all the real numbers except odd multiples of
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pi by 2 and the ranges negative infinity to negative 1 inclusive of negative 1 union of that with
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1 to infinity once again one is included here. Similarly the domain of cos sec function is all
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real numbers except integral multiples of pi. The range is the same as that of secant function.
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So if you make a table you can revise it, you can prepare well for any exam or for that matter
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for making the concept clearer to you. So we have covered today for the domain and range of all
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trigofunctions in a very systematic way. So revise the table regularly so that you remember it for
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your exams very easily and it is very very helpful to recall at any point of time beta.
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So keep on learning my dear friends, don't forget to share this with your friends who are struggling
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with the domain and ranges for the trigumetric functions. Since we are going on with the series
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we will be carrying on with more useful tips in the next episodes and maths is going to be fun
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and simple for you all. Keep learning my dear friends, till the next time.