Sine is one of the basic trigonometric functions, the use of which is not limited to geometry alone. Tables for calculating trigonometric functions, as well as engineering calculators, are not always at hand, and sometimes calculating sine is needed for solving various problems. In general, calculating the sine will help to consolidate drawing skills and knowledge of trigonometric identities.

Games with a ruler and a pencil

A simple task: how to find the sine of an angle drawn on paper? You will need a regular ruler, a triangle (or a compass) and a pencil. The simplest way to calculate the sine of the angle can be by dividing the far leg of the triangle with a right angle on the long side - the hypotenuse. Thus, you first need to supplement the acute angle to the shape of a right-angled triangle, drawing a line perpendicular to one of the rays at an arbitrary distance from the top of the corner. It will be necessary to keep the angle exactly 90 °, for which we will need a stationery triangle.

Using a compass is a little more accurate, but it takes longer. On one of the beams, it is necessary to mark 2 points at a certain distance, adjust the radius approximately equal to the distance between the points on the compass, and draw semicircles with the centers at these points until the intersections of these lines are obtained. Connecting the intersection points of our circles to each other, we get a strict perpendicular to the ray of our angle, it remains only to extend the line to the intersection with another ray.

In the resulting triangle you need to measure with a ruler the side opposite the angle and the long side on one of the rays. The ratio of the first measurement to the second and will be the desired value of the sine of the acute angle.

How to find the sine of the angle

Find the sine for an angle greater than 90 °

For obtuse angle, the task is not much harder. You need to draw a ray from the vertex to the opposite side with a ruler for the formation of a straight line with one ray of light to the needed angle. Obtained with a sharp corner should be done as described above, the sinuses are adjacent angles together form a straight angle of 180°, are equal.

Calculation of sine for other trigonometric functions

Also, sine calculation is possible if the values ​​of other trigonometric functions of the angle or at least the lengths of the sides of the triangle are known. This will help us trigonometric identities. Let us examine common examples.

How to find the sine at the known cosine of the angle? The first trigonometric identity, derived from the Pythagorean theorem, states that the sum of the squares of sine and cosine of the same angle is equal to one.

How to find the sine at a certain tangent of an angle? The tangent is obtained by dividing the far side to the middle or dividing the sine to the cosine. Thus, the sine will be the product of the cosine of the tangent, and the square of the sine is the square of this work. Model the cosine squared of the difference between the unit square and sine according to the first trigonometric identity and by simple manipulations give the equation to calculate the square of the sine, the tangent, respectively, to calculate the sine will have to extract root of the result.

How to find the sine with a certain cotangent of an angle? The cotangent ratio can be calculated by dividing the length of the middle angle leg length distant, and dividing the cosine by the sine, and cotangent is the inverse tangent on the number 1. To calculate sine you can calculate the tangent by the formula tg α = 1 / ctg α and use the formula in the second embodiment. You can also generate a direct formula by analogy with the tangent, which will look like the following.

How to find the sine on the three sides of the triangle

There is a formula for finding the length of the unknown side of any triangle, not only rectangular, on two known sides using the trigonometric cosine function of the opposite angle. It looks like this.

  Well, the sine can be further calculated by the cosine according to the formulas above.