Question

The number of points of intersection of curve sinx = cosy and x² + y² = 1, is-

• A. 3
• B. 2
• C. 0
• D. infinitely many

Answer 1

Answer: (C)

0

Answer 2

We need to solve the system:

$$\sin x = \cos y \quad \text{and} \quad x^2 + y^2 = 1.$$

Step 1. Rewrite the trigonometric equation

Using the identity $\cos y = \sin\left(\frac{\pi}{2}-y\right)$, we have:

$$\sin x = \sin\left(\frac{\pi}{2}-y\right).$$

This implies:

$$\text{(i)}\quad x = \frac{\pi}{2} - y + 2\pi k \quad \text{or} \quad \text{(ii)}\quad x = \pi - \left(\frac{\pi}{2} - y\right) + 2\pi k,$$

where $k \in \mathbb{Z}$.

Simplify (ii):

$$x = \pi - \frac{\pi}{2} + y + 2\pi k = \frac{\pi}{2} + y + 2\pi k.$$

So the two cases are:

$$\text{Case 1: } x = \frac{\pi}{2} - y + 2\pi k,\quad \text{Case 2: } x = \frac{\pi}{2} + y + 2\pi k.$$

Step 2. Substitute in the circle equation

Since the circle $x^2+y^2=1$ restricts $x$ and $y$ to the interval $[-1, 1]$, the shifts $2\pi k$ (for any $k \neq 0$) would lead to values far outside this range. Hence, we consider $k=0$ only.

• For Case 1:

$$x = \frac{\pi}{2} - y.$$

Substitute into the circle:

$$\left(\frac{\pi}{2} - y\right)^2 + y^2 = 1 \quad \Longrightarrow \quad \frac{\pi^2}{4} - \pi y + 2y^2 = 1.$$

This is a quadratic in $y$:

$$2y^2 - \pi y + \left(\frac{\pi^2}{4} - 1\right) = 0.$$

The discriminant $D$ is:

$$D = \pi^2 - 4 \cdot 2 \left(\frac{\pi^2}{4} - 1\right) = \pi^2 - 8\left(\frac{\pi^2}{4} - 1\right) = \pi^2 - 2\pi^2 + 8 = 8 - \pi^2.$$

Since $\pi^2 \approx 9.87$, we have $D < 0$. No real solutions exist.

• For Case 2:

$$x = \frac{\pi}{2} + y.$$

Substitute into the circle:

$$\left(\frac{\pi}{2} + y\right)^2 + y^2 = 1 \quad \Longrightarrow \quad \frac{\pi^2}{4} + \pi y + 2y^2 = 1.$$

The quadratic in $y$ is:

$$2y^2 + \pi y + \left(\frac{\pi^2}{4} - 1\right) = 0.$$

Its discriminant is:

$$D = \pi^2 - 4 \cdot 2 \left(\frac{\pi^2}{4} - 1\right) = 8 - \pi^2,$$

which is also negative.

Thus, in both cases, no real $y$ (and hence no $x$) satisfy the equations simultaneously.

Step 3. Conclusion

There are no intersection points between the curves.