Question

Solve $\sin \left( x \right) + \cos \left( x \right) = 5$.

Answer 1

Hint: The given question involves solving a trigonometric equation and finding the value of angle $x$ that satisfies the given equation. There can be various methods to solve a specific trigonometric equation. For solving such questions, we need to have knowledge of basic trigonometric formulae and identities.

Complete step by step solution:
The given problem requires us to solve the trigonometric equation $\sin \left( x \right) + \cos \left( x \right) = 5$.
The given trigonometric equation can be solved by condensing both the terms into a single compound angle sine or cosine formula.
Dividing both sides of the equation by $\dfrac{1}{{\sqrt 2 }}$ to form a condensed trigonometric formula, we get,
$\dfrac{{\sin \left( x \right)}}{{\sqrt 2 }} + \dfrac{{\cos \left( x \right)}}{{\sqrt 2 }} = \dfrac{5}{{\sqrt 2 }}$
We know that $\sin \left( {\dfrac{\pi }{4}} \right) = \cos \left( {\dfrac{\pi }{4}} \right) = \dfrac{1}{{\sqrt 2 }}$, we get,
$ = \sin \left( x \right)\cos \left( {\dfrac{\pi }{4}} \right) + \cos \left( x \right)\sin \left( {\dfrac{\pi }{4}} \right) = \dfrac{5}{{\sqrt 2 }}$
Using the compound formulae of sine and cosine, $\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B$ and $\cos (A + B) = \cos A\cos B - \sin A\sin B$, we get,
$ = \sin \left( {x + \dfrac{\pi }{4}} \right) = \dfrac{5}{{\sqrt 2 }}$
Now, we know the range of the sine trigonometric function as $\left[ { - 1,1} \right]$. As $\dfrac{5}{{\sqrt 2 }}$ does not lie in the range of the sine function. So, there is no solution for the simplified trigonometric equation $\sin \left( {x + \dfrac{\pi }{4}} \right) = \dfrac{5}{{\sqrt 2 }}$.
Therefore, there exists no solution for the original trigonometric equation $\sin \left( x \right) + \cos \left( x \right) = 5$ also.

Note:
> Such trigonometric equations can be solved by various methods by applying suitable trigonometric identities and formulae.
> The general solution of a given trigonometric solution may differ in form, but actually represents the correct solutions. The different forms of general equations are interconvertible into each other.
> For solving such types of questions where we have to solve trigonometric equations, we need to have basic knowledge of algebraic rules and identities as well as a strong grip on trigonometric formulae and identities.
> We must keep in mind the range and domain of several basic and standard functions to make sure that they are defined at every data point.
> If we see the maximum value of $\sin x$ and $\cos x$ is $1$. So, the maximum value of $\sin x + \cos x$ can take $2$. In this way also, we can say the given statement is invalid.