The number of points of intersection of curve $\sin x = \cos y$ and circle $x^2 + y^2 = 1$.
Step-by-Step Solution
Key Concept: The range of $\cos\theta \pm \sin\theta$ is bounded by $\sqrt{2}$, which cannot equal $\frac{\pi}{2} - 2n\pi$ for any integer $n$.
<h2>Solution: Finding Intersection Points of $\sin x = \cos y$ and $x^2 + y^2 = 1$</h2>
<p>We need to find all points $(x, y)$ that simultaneously satisfy both:</p>
$$\sin x = \cos y \quad \text{...(1)}$$
$$x^2 + y^2 = 1 \quad \text{...(2)}$$
<h3>Step 1: Analyze the constraint from the circle</h3>
<p>From equation (2), the point $(x, y)$ lies on the unit circle. Therefore:</p>
$$-1 \le x \le 1 \quad \text{and} \quad -1 \le y \le 1$$
<h3>Step 2: Use the Pythagorean identity</h3>
<p>Since $(x, y)$ is on the unit circle, we can write:</p>
$$x = \cos \theta \quad \text{and} \quad y = \sin \theta$$
<p>for some real number $\theta$.</p>
<h3>Step 3: Substitute into the curve equation</h3>
<p>Substituting the parametrization into equation (1):</p>
$$\sin(\cos \theta) = \cos(\sin \theta) \quad \text{...(3)}$$
<h3>Step 4: Analyze the range constraints</h3>
<p>For any $\theta \in \mathbb{R}$:</p>
$$\cos \theta \in [-1, 1] \quad \Rightarrow \quad \sin(\cos \theta) \in [\sin(-1), \sin(1)]$$
<p>Since $\sin$ is increasing on $[-1, 1]$, we have:</p>
$$\sin(\cos \theta) \in [-\sin(1), \sin(1)]$$
<p>Similarly:</p>
$$\sin \theta \in [-1, 1] \quad \Rightarrow \quad \cos(\sin \theta) \in [\cos(1), 1]$$
<p>Since $\cos$ is decreasing on $[0, 1]$ and even:</p>
$$\cos(\sin \theta) \in [\cos(1), 1]$$
<h3>Step 5: Compare the ranges</h3>
<p>We now have:</p>
$$\sin(\cos \theta) \in [-\sin(1), \sin(1)]$$
$$\cos(\sin \theta) \in [\cos(1), 1]$$
<p>Let us evaluate the numerical bounds. We know:</p>
$$\sin(1) \approx 0.8414 \quad \text{and} \quad \cos(1) \approx 0.5403$$
<p>Therefore:</p>
$$\sin(\cos \theta) \in [-0.8414, 0.8414]$$
$$\cos(\sin \theta) \in [0.5403, 1]$$
<h3>Step 6: Determine the overlap</h3>
<p>The ranges overlap in the interval $[0.5403, 0.8414]$. However, we must check more carefully whether equation (3) can be satisfied.</p>
<h3>Step 7: Rigorous analysis using calculus</h3>
<p>Consider the function:</p>
$$f(\theta) = \sin(\cos \theta) - \cos(\sin \theta)$$
<p>At $\theta = 0$:</p>
$$f(0) = \sin(1) - \cos(0) = \sin(1) - 1 \approx 0.8414 - 1 = -0.1586 < 0$$
<p>At $\theta = \pi/2$:</p>
$$f(\pi/2) = \sin(0) - \cos(1) = 0 - \cos(1) \approx -0.5403 < 0$$
<p>At $\theta = \pi$:</p>
$$f(\
Correct Answer: 0