Chapter 29: Chapter 29

Problem 920

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

Short Answer

Expert verified

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

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Step by step solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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.

Step 6: Answer

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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Chapter 29: Chapter 29

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

Short Answer

Step by step solution

Unlock all solutions

Step 1: Rewrite the equation using trigonometric identities

Step 2: Use the cosine property to find solutions

Step 3: Solve the first equation

Step 4: Solve the second equation

Step 5: Find the solutions in the given interval

Step 6: Answer

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