Question

Find all solutions in the interval [0, 2π).

cos x = sin x

Help me please

Answer 1

Answer on Question #42395 – Math – Trigonometry

Find all solutions in the interval $[0, 2\pi)$.

$$\cos x = \sin x$$

Help me please

Solution

$$\cos x = \sin x \rightarrow \frac{\cos x}{\cos x} = \frac{\sin x}{\cos x} \rightarrow \tan x = 1 \rightarrow x = \frac{\pi}{4} + \pi n, \quad n \in \mathbb{Z}.$$

We can divide by $\cos x \neq 0$, because $\cos x$ and $\sin x$ cannot equals zero simultaneously due to equality $\sin^2 x + \cos^2 x = 1$.

Therefore, in the interval $[0, 2\pi]$ the equation $\cos x = \sin x$ has two solutions:

$$x = \frac{\pi}{4} \quad \text{and} \quad x = \frac{\pi}{4} + \pi = \frac{5\pi}{4}.$$

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