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Problem 920

# Find the solution set on $$(0,2 \pi)$$ for $$\sin \mathrm{x}=\cos \mathrm{x}$$.

Expert verified
The solution set for the equation $$\sin{x} = \cos{x}$$ on the interval $$(0, 2\pi)$$ is $$\left\{\frac{\pi}{4}, \frac{\pi}{2}, \frac{5\pi}{4}\right\}$$.
See the step by step solution

## Step 1: Rewrite the equation using trigonometric identities

To rewrite the equation $$\sin{x} = \cos{x}$$, we can use the identity $$\cos(\frac{\pi}{2} - x) = \sin{x}$$. The equation then becomes: $$\cos\left(\frac{\pi}{2} - x\right) = \cos{x}.$$

## Step 2: Use the cosine property to find solutions

The equation, $$\cos\left(\frac{\pi}{2} - x\right) = \cos{x}$$ implies that either $$x = \frac{\pi}{2} - x + 2n\pi$$ or $$x = - \left(\frac{\pi}{2} - x\right) + 2n\pi$$ for some integer $$n$$. Let's now solve these equations separately.

## Step 3: Solve the first equation

Now we will solve the equation $$x = \frac{\pi}{2} - x + 2n\pi$$: $$2x = \frac{\pi}{2} + 2n\pi$$ $$x = \frac{\pi}{4} + n\pi .$$

## Step 4: Solve the second equation

Now we will solve the equation $$x = - \left(\frac{\pi}{2} - x\right) + 2n\pi$$: $$2x = \pi - 2n\pi$$ $$x = \frac{\pi}{2} - n\pi .$$ Finally, we can notice that if $$n$$ is odd, the second equation is the same as the first equation.

## Step 5: Find the solutions in the given interval

We have the general solutions $$x = \frac{\pi}{4} + n\pi$$ and $$x = \frac{\pi}{2} - n\pi$$. Now we will find the solutions in the interval $$(0, 2\pi)$$: For $$x = \frac{\pi}{4} + n\pi$$: - For $$n = 0$$, $$x = \frac{\pi}{4}$$ which lies in the interval. - For $$n = 1$$, $$x = \frac{5\pi}{4}$$ which lies in the interval. - For $$n = 2$$, $$x = \frac{9\pi}{4}$$ which does not lie in the interval. For $$x = \frac{\pi}{2} - n\pi$$: - For $$n = 0$$, $$x = \frac{\pi}{2}$$ which lies in the interval. - For $$n = 1$$, $$x = -\frac{\pi}{2}$$ which does not lie in the interval.

The solution set for the equation $$\sin{x} = \cos{x}$$ on the interval $$(0, 2\pi)$$ is $$\left\{\frac{\pi}{4}, \frac{\pi}{2}, \frac{5\pi}{4}\right\}$$.

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