Sine is one of the main trigonometric functions, the application of which is not limited to just geometry. Tables for calculating trigonometric functions, like engineering calculators, are not always at hand, and calculation of the sine is sometimes necessary to solve various problems. In general, calculating the sine will help to consolidate the drawing skills and knowledge of trigonometric identities.

Games with ruler and pencil

A simple task: how to find the sine of a corner drawn on paper? For the solution you need a regular ruler, a triangle (or compass) and a pencil. The simplest way to calculate the sine of an angle is by dividing the far corner of the triangle with a right angle to the long side - the hypotenuse. Thus, first we need to add an acute angle to the figure of a rectangular triangle by drawing a line perpendicular to one of the rays at an arbitrary distance from the vertex of the corner. It will be required to keep the angle exactly 90 °, for which we will need a writing triangle.

Using the compass is a bit more precise, but it takes more time. On one of the beams, you should note 2 points at some distance, adjust the radius on the compass approximately equal to the distance between the points, and draw semicircles with the centers at these points until the intersections of these lines are obtained. By connecting the intersection points of our circles with each other, we get a strict perpendicular to the ray of our angle, it remains only to extend the line until it intersects with another ray.

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

How to find the sine of an angle

Find a 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.

Calculating the sine by other trigonometric functions

Also, the calculation of the sine 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. Trigonometric identities help us in this. Let us examine the common examples.

How to find the sine with the known cosine of the angle? The first trigonometric identity, starting from the Pythagorean theorem, states that the sum of the squares of the 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 a sine on three sides of a triangle

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

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