Animated Solution for Mathematics - Trigonometry: The general solution of the trigonometric equation $\sin x+\cos x=1$ is given by :

Welcome, future engineers! Today, we are going to peel back the layers of a seemingly simple trigonometric equation: $\sin x+\cos x=1$.

Many students see this and immediately reach for their algebraic toolkit, but I want you to pause. Before we touch a single variable, let us visualize the reality behind these functions.

Imagine we define two new variables: $X=\cos x$ and $Y=\sin x$. We know the fundamental identity of trigonometry, the heartbeat of the unit circle:

This is the equation of a circle centered at the origin with a radius of one. Now, look at our original equation: $\sin x+\cos x=1$. In our new coordinate system, this is simply $Y+X=1$, or $X+Y=1$.

This is the equation of a straight line! The solutions to our trigonometric equation are not just abstract numbers; they are the physical intersection points where this line $X+Y=1$ cuts through our unit circle.

If you sketch this, you will see the line passing through $(1,0)$ and $(0,1)$. These are our principal solutions: $x=0$ and $x=\frac{\pi }{2}$.

But we are not just looking for a snapshot; we are looking for the general solution, the infinite set of all possible rotations.

Now that we have our geometric intuition, let us formalize it. Whenever you encounter an equation of the form $a\sin x+b\cos x=c$, there is a powerful, standard technique that will save you hours of frustration.

We divide the entire equation by $\sqrt{a^2+b^2}$. In our case, $a=1$ and $b=1$, so our divisor is $\sqrt{1^2+1^2}=\sqrt{2}$.

When we divide every term by $\sqrt{2}$, we get:

Why do we do this? Because $\frac{1}{\sqrt{2}}$ is a special value. We know that $\cos (\frac{\pi }{4})=\frac{1}{\sqrt{2}}$ and $\sin (\frac{\pi }{4})=\frac{1}{\sqrt{2}}$.

Substituting these, our equation transforms into:

Look at the left side of that equation. It is the classic expansion of the sine addition formula: $\sin (A+B)=\sin A\cos B+\cos A\sin B$.

By recognizing this, we compress the entire expression into a single, beautiful term:

We have reduced a complex sum of two functions into a single sine function. Now, we know that $\frac{1}{\sqrt{2}}$ is also $\sin (\frac{\pi }{4})$.

So, our equation is simply $\sin (x+\frac{\pi }{4})=\sin (\frac{\pi }{4})$. This is the perfect form for the general solution formula: $\sin \theta =\sin \alpha$, which implies $\theta =n\pi +(-1{)}^n\alpha$, where $n$ is any integer.

Substituting our values, we get:

Finally, to isolate $x$, we subtract $\frac{\pi }{4}$ from both sides:

This expression is the key that unlocks every single solution, for every rotation, for every integer $n$. It is not just an answer; it is the mathematical description of the infinite dance of the sine and cosine functions.

Keep practicing this method, and you will find that even the most intimidating trigonometric equations become simple, elegant, and deeply satisfying to solve.