75. Parametric equations of a line are given: $x=R \cos t$, $y=R \sin t$. Convert them to an equation with two variables.
Solution. From the second equation

![](https://cdn.mathpix.com/cropped/2024_05_22_db4d450c77c65a914ec1g-131.jpg?height=354&width=423&top_left_y=1444&top_left_x=115)

Fig. 44, we express $\sin t=\frac{y}{R}$. Knowing that $\cos t=$ $=\sqrt{1-\sin ^{2} t}$, we find $\cos t=\sqrt{1-\frac{y^{2}}{R^{2}}}$.

Now substitute this expression into the first equation:

$$
\begin{gathered}
x=R \sqrt{1-\frac{y^{2}}{R^{2}}} ; x=R \sqrt{\frac{R^{2}-y^{2}}{R^{2}}} \\
x=\sqrt{R^{2}-y^{2}}, \text { i.e. } x^{2}+y^{2}=R^{2}
\end{gathered}
$$

We have obtained the well-known equation of a circle with radius $R$ and center at the origin from the school course.