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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 \(\left\{\frac{\pi}{4}, \frac{\pi}{2}, \frac{5\pi}{4}\right\}\).

Step by step solution

01

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: $$ \cos\left(\frac{\pi}{2} - x\right) = \cos{x}. $$
02

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.
03

Solve the first equation

Now we will solve the equation \(x = \frac{\pi}{2} - x + 2n\pi\): $$ 2x = \frac{\pi}{2} + 2n\pi $$ $$ x = \frac{\pi}{4} + n\pi . $$
04

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.
05

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.
06

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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Most popular questions from this chapter

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