Skip to main content

Trigonometry Equation - Notes, Concept and All Important Formula

TRIGONOMETRIC EQUATION

1. TRIGONOMETRIC EQUATION :

An equation involving one or more trigonometrical ratios of unknown angles is called a trigonometric equation.


2. SOLUTION OF TRIGONOMETRIC EQUATION :

A value of the unknown angle which satisfies the given equations is called a solution of the trigonometric equation.

(a) Principal solution :- The solution of the trigonometric equation lying in the interval \([0,2 \pi]\).

(b) General solution :- Since all the trigonometric functions are many one & periodic, hence there are infinite values of \(\theta\) for which trigonometric functions have the same value. All such possible values of \(\theta\) for which the given trigonometric function is satisfied is given by a general formula. Such a general formula is called general solutions of trigonometric equation.


3. GENERAL SOLUTIONS OF SOME TRIGONOMETRICE EQUATIONS (TO BE REMEMBERED) :

  (a) If \(\sin \theta=0\), then \(\theta= n \pi, n \in I\) (set of integers)

  (b) If \(\cos \theta=0\), then \(\theta=(2 n+1) \frac{\pi}{2}, n \in I\)

  (c) If \(\tan \theta=0\), then \(\theta= n \pi, n \in I\)

  (d) If \(\sin \theta=\sin \alpha\), then \(\theta= n \pi+(-1)^{n} \alpha, n \in I\)

  (e) If \(\cos \theta=\cos \alpha\), then \(\theta=2 n \pi \pm \alpha, n \in I\)

  (f) If \(\tan \theta=\tan \alpha\), then \(\theta= n \pi+\alpha, n \in I\)

  (g) If \(\sin \theta=1\), then \(\theta=2 n \pi+\frac{\pi}{2}=(4 n +1) \frac{\pi}{2}, n \in I\)

  (h) If \(\cos \theta=1\) then \(\theta=2 n \pi, n \in I\)

  (i) If \(\sin ^{2} \theta=\sin ^{2} \alpha\) or \(\cos ^{2} \theta=\cos ^{2} \alpha\) or \(\tan ^{2} \theta=\tan ^{2} \alpha\)  then  \(\theta= n \pi \pm \alpha, n \in I\)

  (j) \(\sin ( n \pi+\theta)=(-1)^{ n } \sin \theta, n \in I\)

       \(\cos ( n \pi+\theta)=(-1)^{ n } \cos \theta, n \in I\)


4. GENERAL SOLUTION OF EQUATION a \(\sin \theta+ b \cos \theta= c\) :

Consider, a \(\sin \theta+b \cos \theta=c \ldots \ldots \ldots \ldots \ldots\) (i)

\(\therefore \frac{a}{\sqrt{a^{2}+b^{2}}} \sin \theta+\frac{b}{\sqrt{a^{2}+b^{2}}} \cos \theta=\frac{c}{\sqrt{a^{2}+b^{2}}}\)

equation (i) has the solution only if \(|c| \leq \sqrt{a^{2}+b^{2}}\)

let \(\frac{ a }{\sqrt{ a ^{2}+ b ^{2}}}=\cos \phi, \frac{ b }{\sqrt{ a ^{2}+ b ^{2}}}=\sin \phi \quad \& \phi=\tan ^{-1} \frac{ b }{ a }\)

by introducing this auxiliary argument \(\phi\), equation (i) reduces to

\((\theta+\phi)=\frac{c}{\sqrt{a^{2}+b^{2}}}\)

Now this equation can be solved easily.


5. GENERAL SOLUTION OF EQUATION OF FORM :

\(a_{0} \sin ^{n} x+a_{1} \sin ^{n-1} x \cos x+a_{2} \sin ^{n-2} x \cos ^{2} x+\ldots \)\(\ldots \ldots .+a_{n} \cos ^{n} x=0\)

\(a_{0}, a_{1}, \ldots \ldots . a_{n}\) are real numbers

Such an equation is solved by dividing equation both sides by \(\cos ^{n} x\).


6. IMPORTANT TIPS:

  (a) For equations of the type \(\sin \theta= k\) or \(\cos \theta= k\), one must check that \(| k | \leq 1\)

  (b) Avoid squaring the equations, if possible, because it may lead to extraneous solutions.

  (c) Do not cancel the common variable factor from the two sides of the equations which are in a product because we may loose some solutions.

  (d) The answer should not contain such values of \(\theta\), which make any of the terms undefined or infinite.

  (e) Check that denominator is not zero at any stage while solving equations.

  (f) (i) If \(\tan \theta\) or \(\sec \theta\) is involved in the equations, \(\theta\) should not be odd multiple of \(\frac{\pi}{2}\).

       (ii) If \(\cot \theta\) or \(\operatorname{cosec} \theta\) is involved in the equation, \(\theta\) should not be integral multiple of \(\pi\) or 0 .

  (g) If two different trigonometric ratios such as \(\tan \theta\) and \(\sec \theta\) are involved then after solving we cannot apply the usual formulae for general solution because periodicity of the functions are not same.

  (h) If L.H.S. of the given trigonometric equation is always less than or equal to \(k\) and \(RHS\) is always greater than \(k\), then no solution exists. If both the sides are equal to \(k\) for same value of \(\theta\), then solution exists and if they are equal for different value of \(\theta\), then solution does not exist.

Labels:

Comments

Post a Comment

Popular posts from this blog

Function (Mathematics)- Notes, Concept and All Important Formula

FUNCTION 1. DEFINITION : If to every value (considered as real unless other-wise stated) of a variable \(\mathrm{x}\) , which belongs to a set \(\mathrm{A}\) , there corresponds one and only one finite value of the quantity \(y\) which belong to set \(B\) , then \(y\) is said to be a function of \(x\) and written as \(f: A \rightarrow B, y=f(x), x\) is called argument or independent variable and \(y\) is called dependent variable. Pictorially: \( \underset{\text { input }}{\stackrel{\mathrm{x}}{\longrightarrow}}\boxed{\mathrm{f}}\,\,  \underset{\text { output }}{\stackrel{\mathrm{f(x)=y}}{\longrightarrow}}\) \(y\) is called the image of \(x\) and \(x\) is the pre-image of \(y\) , under mapping \(\mathrm{f}\) .  Every function \(\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}\) satisfies the following conditions. (i) \(f \subset A \times B\) (ii) \(\forall \,\,  a \in A \,\, \exists\,\, b \in B\) such that \((a, b) \in f\) and (iii) If \((a, b) \in f \) &...
Post a Comment
Read more

Statistics - Notes, Concept and All Important Formula

STATISTICS MEASURES OF CENTRAL TENDENCY: An average value or a central value of a distribution is the value of variable which is representative of the entire distribution, this representative value are called the measures of central tendency. Generally there are following five measures of central tendency: (a) Mathematical average (i) Arithmetic mean.           (ii) Geometric mean.           (iii) Harmonic mean (b) Positional average (i) Median.       (ii) Mode All Chapter Notes, Concept and Important Formula 1. ARITHMETIC MEAN/MEAN: (i) For ungrouped dist.: If \(x _{1}, x _{2}, \ldots \ldots x _{ n }\) are \(n\) values of variate \(x\) then their mean \(\bar{x}\) is defined as \(\overline{ x }=\dfrac{ x _{1}+ x _{2}+\ldots . .+ x _{ n }}{ n }=\frac{\displaystyle \sum_{ i =1}^{ n } x _{ i }}{ n }\) \(\Rightarrow\displaystyle \Sigma x _{ i }= n \overline{ x }\) (ii) For ungrouped and grouped freq. dist.:...
Post a Comment
Read more

What are Function and how its work on Calculus?

What are Function ? Table Of Contents Introduction with beautiful example Here's a plant, and what you see here is it's  shadow. Can you list the things that the length of the shadow is dependent on. One, it's dependent on the position of the source of light. Anything else that you can think of. If the height of the plant grows then the shadows length will also change, right. So the length of the shadow is dependent on the position of the source of light, and the height of the plant too. So we can say that the length of the shadow is a function of the following two things. The output is dependent on these two things, which could be considered as the inputs. That's a very simple way to understand functions. Could you think of more inputs, this output is dependent on here, tell us yours answers in the comment section below.  How do Function work in calculus ? That's what we'll see in this topics. Previously, we saw an idea to find the instant...
1 comment
Read more