761. It is known that $\operatorname{tg} \alpha=2$ (for the acute angle $\alpha$). Find $\sin \alpha$ and $\cos \alpha$.
$\triangleright$ Let's take the adjacent side as a unit, then the opposite side will be 2. By the Pythagorean theorem, the hypotenuse is $\sqrt{1^{2}+2^{2}}=\sqrt{5}$, so $\sin \alpha=2 / \sqrt{5}$ and $\cos \alpha=1 / \sqrt{5} . \triangleleft$

In general: if $\operatorname{tg} \alpha=a$, then the hypotenuse will be $\sqrt{1+a^{2}}$, so $\cos \alpha=1 / \sqrt{1+a^{2}}$ and $\sin \alpha=a / \sqrt{1+a^{2}}$. In other words, we can write such formulas expressing the sine and cosine of an acute angle through its tangent:

$$
\cos \alpha=\frac{1}{\sqrt{1+\operatorname{tg}^{2} \alpha}} ; \quad \sin \alpha=\frac{\operatorname{tg} \alpha}{\sqrt{1+\operatorname{tg}^{2} \alpha}}
$$

Expressing the tangent through the sine (or cosine) is even simpler: since $\cos ^{2} \alpha+\sin ^{2} \alpha=1$, then

$$
\cos \alpha=\sqrt{1-\sin ^{2} \alpha}
$$

so,

$$
\operatorname{tg} \alpha=\frac{\sin \alpha}{\sqrt{1-\sin ^{2} \alpha}}
$$

(similarly for cosine).

These formulas allow finding the sine from the tangent or vice versa, but how do you find the sine if the degree measure of the angle is known? For this, books of tables were published. For example, in the USSR, the book "Eight-digit tables of trigonometric functions" by L. S. Khrenov was repeatedly published (464 pages in the 1967 edition); his "Seven-digit tables..." occupied 406 pages (1971 edition), "Six-digit tables..." of the 1978 edition - 367 pages, and "Five-digit tables..." (1949 edition) - only 103 pages. In the standard set of school textbooks, the "tables of Bradis" - a brochure with various mathematical tables compiled by V. M. Bradis - have long been included. A fragment of the sine table from this brochure is reproduced in the image above. (Of course, the values in these tables were known even before Khrenov and Bradis; they only organized the material and prepared the publication.) Amazingly, these tables are still being reprinted, and in online stores, you can buy an edition from 2012, which, like many years ago, opens with the table "Exact products of two-digit numbers."

The definition of sine and cosine can be restated in other words.

Take a point on a circle of unit radius and let this point be visible from the center of the circle at an angle $\alpha$ to the horizontal (see the figure). Then the coordinates of this point will be $(\cos \alpha, \sin \alpha)$.

![](https://cdn.mathpix.com/cropped/2024_05_21_90703b5d5e76e3b5cd3dg-226.jpg?height=389&width=585&top_left_y=1847&top_left_x=610)

![](https://cdn.mathpix.com/cropped/2024_05_21_90703b5d5e76e3b5cd3dg-227.jpg?height=274&width=343&top_left_y=820&top_left_x=351)

$(\cos \alpha, \sin \alpha)$
![](https://cdn.mathpix.com/cropped/2024_05_21_90703b5d5e76e3b5cd3dg-227.jpg?height=752&width=350&top_left_y=1481&top_left_x=358)