Question

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

Answer 1

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.