How to find the angle of the known sine

In everyday life, to solve trigonometric problems, one must know how to find the angle over a known sine. It turns out that this is not difficult at all. To do this, remember that sin150 = 0,26, sin300 = 0,5, sin450 = 0,707 and sin900 = 1, which are known to everyone from school geometry. In addition, for intervals from 150 to 300 and from 300 to 450 it is necessary to know that sinuses can be calculated as follows: the difference between sin300 and sin150 is 0,50 – 0,26 = 0,24, and sin450 - sin300 = 0,707 – 0,5 = 0,207. Hence, with an increase in the angle per degree, the sine grows by about 1/15 the fraction of this difference, i.e. for intervals from 150 to 300 and from 300 to 450 we have 0,24/15 = 0,016 and 0,207/15 = 0,014, respectively. To find the sines of angles in these intervals, these values are added sequentially to sin150, or sin300, respectively. Strictly speaking, this is not so, but a deviation from this rule is found only in the third significant figure, which in everyday life can be discarded.

In everyday life, to solve trigonometric problems, one must know how to find the angle over a known sine

Knowing this, let's try to find the angle whose sine is known and equal to 0,38. Since 0,38 is less than 0,5, the required angle is less than 300. But it is greater than 150, since sin150 is equal to 0,26. To find it in the interval between 150 and 300, we compose the proportion:

õ : 30 = 0,38 : 0,5

Consequently:

x = 22,80, or approximately 230.

Another example: find the angle whose sine is 0,62.

Since 0,62 is between 0,5 and 0,707, the angle should be in the interval between 300 and 450. We form the proportion:

õ : 30 = 0,62 : 0,5,

from where:

x = 37,20, or approximately 370.

Finally, the third example: find the angle whose sine is 0,91.

Since 0,91 is between 0,71 and 1, the angle lies in the interval between 450 and 900. In the drawing BC there is a sine A if AB = 1. Knowing BC, it is easy to find the sine B.

ÀÑ2 = 1 – ÂÑ2 = 1 - 0,912 = 1 – 0,8281 = 0,1729, ÀÑ = quad. square 0,1729 = 0,42.

Now we find the value B, whose sine is equal to 0,42; after that it will be easy to find the angle A, equal to 900 - B. Since 0,42 is between 0,26 and 0,5, B lies in the interval between 150 and 300. It is determined from the proportion:

 : 30 = 0,42 : 0,5
 = 25,20

Hence, A = 900 - B = 900 - 25,20 = 64,80, or with a rounding of 650.

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