Trigonometric Equations

Trigonometric Equation

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

Solution/Roots of a Trigonometric Equation

A value of the unknown angle which satisfies the given equation is called a solution or root of the equation.

The trigonometric equation may have infinite number of solutions.

1. Principal Solution: The least value of unknown angle which satisfies the given equation, is called a principal solution of trigonometric
2. General Solution: We know that, trigonometric function are periodic and s olution of trigonometric equations can be generalised with the help of the periodicity of the trigonometric functions. The solution consistring of all possible solutions of a trigonometric equation is called its general solution.

Important Results

1. $\sin \theta =0\Rightarrow \theta =n\pi$
2. $\cos \theta =0\Rightarrow \theta =\left(2n+1\right)\frac{\pi }{2}$
3. $\tan \theta =0\Rightarrow \theta =n\pi$
4. $\sin \theta =\sin \alpha \Rightarrow \theta =n\pi +{\left(-1\right)}^n\alpha ,$ where $\alpha \in \left[-\frac{\pi }{2},\frac{\pi }{2}\right]$
5. $\cos \theta =\cos \alpha \Rightarrow \theta =2n\pi \pm \alpha ,$ where $\alpha \in \left[0,\pi \right]$
6. $\tan \theta =\tan \alpha \Rightarrow \theta =n\pi +\alpha ,$where $\alpha \in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$
7. ${\sin }^2\theta ={\sin }^2\alpha ,{\cos }^2\theta ={\cos }^2\alpha ,{\tan }^2\theta ={\tan }^2\alpha$ $\Rightarrow \theta =n\pi \pm \alpha$
8. $\sin \theta =1\Rightarrow \theta =\left(4n+1\right)\frac{\pi }{2}$
9. $\cos \theta =1\Rightarrow \theta =2n\pi$
10. $\cos \theta =-1\Rightarrow \theta =\left(2n+1\right)\pi$
11. $\begin{array}{r}\sin \theta =\sin \alpha and\cos \theta =\cos \alpha \\ \sin \theta =\sin \alpha and\tan \theta =\cos \alpha \\ \tan \theta =\tan \alpha and\cos \theta =\cos \alpha \end{array}\}\Rightarrow \theta =2n\pi +\alpha$
12. Equation of the form $\alpha \cos \theta +b\sin \theta =c$

Put $a=r\cos \alpha$ and $b=r\sin \alpha ,$ where$r=\sqrt{a^2+b^2}$ and$\left|c\right|\le \sqrt{a^2+b^2}$

$\therefore \theta =2n\pi \pm \alpha +\varphi ,n\in I$Where;;$\alpha ={\cos }^{-1}\frac{\left|c\right|}{\sqrt{a^2+b^2}}$ and;$\varphi ={\tan }^{-1}\frac{b}{a}$

1. If $\left|c\right|>\sqrt{a^2+b^2,}$ equation has no solution.2. $\left|c\right|\le \sqrt{a^2+b^2,}$;;equation is ;solvable.13. $\sin \left(\frac{n\pi }{2}+\theta ;\right)={\left(-1\right)}^{\frac{n}{2}}\cos \theta ,$ if $n$ is odd.;;;;;;;;;;;;;$={\left(-1\right)}^{\frac{n}{2}}\sin \theta ,$if $n$ is even.14. $\cos \left(\frac{n\pi }{2}+\theta ;\right)={\left(-1\right)}^{\frac{n-1}{2}}\sin \theta ,$if $n$is odd$={\left(-1\right)}^{\frac{n}{2}}\cos \theta ,$if$n$is even.15. $\sin {\theta }_1+\sin {\theta }_2+.......+sin{\theta }_n=n\Rightarrow \sin {\theta }_1=\sin {\theta }_2=.....=\sin {\theta }_n=1$16. $\cos {\theta }_1+\cos {\theta }_2+......+\cos {\theta }_n=n\Rightarrow \cos {\theta }_1=\cos {\theta }_2=.....\cos {\theta }_n=1$17. $\sin \theta +\cos ec\theta =2\Rightarrow \sin \theta =1$18. $\cos \theta +\sec \theta =2\Rightarrow \cos \theta =1$19. $\sin \theta +\cos ec\theta =-2\Rightarrow \sin \theta =-1$20. $\cos \theta +\sec \theta =-2\Rightarrow \cos \theta =-1$Key Points

1. While solvingan equation, we have to square it, sometimes the resulting

1. Do not cancel common factors involving the unknown angle on LHS andRHS. Because it may be the solution of given equation.
2. (a)Equation involving $\sec \theta$or $\tan \theta$ can never be a solution of the form $\left(2n+1\right)\frac{\pi }{2}.$

Hyperbolic Functions

The hyperbolic functions $\sin hz,coshz,tanhz,cosechz,sechz,cothz$ are angles of the circular functions, defined by removing is appearing in the complex exponentials.

1. $\sinh x=\frac{e^x-e^{-x}}{2}$
2. $\cosh x=\frac{e^x+e^{-x}}{2}$
3. $\tanh x=\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}$
4. $\cos echx=\frac{1}{\sinh x}=\frac{2}{e^x-e^{-x}}$
5. $\mathrm{sech}x=\frac{1}{\cosh x}=\frac{2}{e^x+e^{-x}}$
6. $\coth x=\frac{\cosh x}{\sinh x}=\frac{e^x+e^{-x}}{e^x-e^{-x}}$

Identities

1. ${\cosh }^{2}x-{\sinh }^{2}x=1$
2. $\sec h^{2}x+{\tanh }^{2}x=1$
3. ${\coth }^{2}x-\cos ec{h}^{2}x=1$
4. ${\cosh }^{2}x+{\sinh }^{2}x=\cosh 2x$

Formulae for the Sum and Difference

1. $\sinh \left(x\pm y\right)=\sinh x\cosh x\pm \cosh x\sinh y$
2. $\cosh \left(x\pm y\right)=\cosh x\cosh y\pm \sinh x\sinh y$
3. $\tanh \left(x\pm y\right)=\frac{\tanh x\pm \tanh y}{1\pm \tanh x\tanh y}$

Formulae of Transform the Product into sum or Difference

1. $\sinh x+\sinh y=2\sinh \left(\frac{x+y}{2}\right)\cosh \left(\frac{x-y}{2}\right)$
2. $\sinh x-\sinh y=2\cosh \left(\frac{x+y}{2}\right)\sinh \left(\frac{x-y}{2}\right)$
3. $\cosh x+\cosh y=2\cosh \left(\frac{x+y}{2}\right)\cosh \left(\frac{x-y}{2}\right)$
4. $\cosh x-\cosh y=2\sinh \left(\frac{x+y}{2}\right)\sinh \left(\frac{x-y}{2}\right)$
5. $2\sinh x\cosh y=\sinh \left(x+y\right)-\sinh \left(x-y\right)$
6. $2\cosh x\sinh y=\sinh \left(x+y\right)-\sinh \left(x-y\right)$
7. $2\cosh x\cosh y=\cosh \left(x+y\right)+\cosh \left(x-y\right)$
8. $2\sinh x\sinh y=\cosh \left(x+y\right)-\cosh \left(x-y\right)$

Formulae for multiples of X

1. $\sinh 2x=2\sinh x\cosh x=\frac{2\tanh x}{1-{\tanh }^{2}x}$

2. $\cosh 2x={\cosh }^{2}x+{\sinh }^{2}x$

3. $\cosh 2x=2{\cosh }^{2}x-1=1+2{\sinh }^{2}x$

4. $\cosh 2x=\frac{1+{\tanh }^{2}x}{1-{\tanh }^{2}x}$

5. $\tanh 2x=\frac{2\tanh x}{1+{\tanh }^{2}x}$

6. $\sinh 3x=3\sinh x+4{\sinh }^{3}x$

7. $\cosh 3x=4{\cosh }^{3}x-\cosh x$

8. $\tanh 3x=\frac{3\tanh x+{\tanh }^{3}x}{1+3{\tanh }^{2}x}$