Table values for sine 45, cosine 45, and also 45 degree tangent are offered below. The following is an explanation of the method and the correctness the the calculate of these worths for one arbitrary rectangular triangle.

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VALUE of TRIGONOMETRIC features AT α = 45 °

Let"s create and consider a rectangular triangle abc in i m sorry the edge ∠ B = 45 °. Top top the basis of the proportion of that sides, us calculate the worth of trigonometric functions in a rectangular triangle because that an edge of 45 degrees. Since the triangle is rectangular, the worths ​​of the sine, cosine and tangent functions will be equal to the proportion of its respective parties.

since the values ​​of sinus, cosine and tangent features depend exclusively on the degree angle the the angle (or the value expressed in radians), the ratios discovered by us and will be the values ​​of the sinus 45, cosine 45, and the tangent 45 degrees.


follow to the properties of a rectangle-shaped triangle, the edge C is straight and also is same to 90 degrees. Angle B we originally built with a degree degree of 45 degrees. We find the value of the edge A. Since the amount of the angles of the triangle is 180 degrees, then

∠А +∠В +∠С = 180° The angle C is straight and equal to 90 degrees, edge B we first defined together 45 degrees, thus: ∠А = 180° -∠С -∠В = 180° - 90° - 45° = 45°,

because in this triangle two angles room equal, climate the triangle alphabet is a rectangular, and, in ~ the exact same time, one isosceles one, in i beg your pardon both cateches space equal: AC = BC.

Assume the the size of the sides is equal to some variety of AS = BC = a. knowing the size of the coils, we calculate the size of the hypotenuse.

follow to Pythagoras"s theorem: АВ2=АС2+ВС2

Let"s replace the lengths AC and also BC with the change a, then us obtain:

АВ22+ а2= 2а2,

then AB = a√2.

together a result, us expressed the lengths of all sides that a rectangular triangle v an edge of 45 levels through the change a.

follow to the nature of trigonometric attributes in a rectangular triangle, the proportion of the equivalent sides of the triangle will certainly be same to the worth of the matching functions. For this reason for an edge α = 45 degrees:

sin α = BC / AB (according to the an interpretation of a sinus because that a rectangle-shaped triangle - this is the ratio of the opposite angle to the hypotenuse, BC - catheter, abdominal - hypotenuse)

cos α = AC / AB (according to the cosine definition - this is the proportion of the adjacent catheter to the hypotenuse, AC - catheter, abdominal - hypotenuse)

tg α = BC / AC (Similarly, the tangent for the angle α will certainly be equal to the ratio of the opposing catheter to the adjacent)

rather of designating the sides, we will substitute the values ​​of your lengths v the change a.

Proceeding indigenous this (see table of values ​​sin 45, cos 45, tg 45) us obtain:

The table worths ​​sin 45, cos 45, tg 45 (that is, the values ​​of sinus 45, cosine 45, and also 45 level tangent deserve to be calculated as the ratio of the corresponding sides that this triangle), us substitute the calculated values ​​of the lengths of the political parties in the formula and get the an outbarisalcity.orge in the image below.

Table values: sinus 45, cosine 45 and tangent 45 degrees



45 degree tangent equates to one the sinus of 45 degrees is same to the cosine that 45 degrees and is equal to the source of the two halves (the exact same as the unit divisible by the source of the two)

As can be viewed from the calculations given above, to calculation the worths of the matching trigonometric function, no the lengths of the political parties of the triangle room important, yet their ratio, i beg your pardon is constantly the very same for the exact same angles, regardless of the dimension of a details triangle.

Sine, cosine and tangent angle π / 4 radians

In the work proposed for addressing in an elderly classes and tests, instead of the degree angle that the edge is often uncovered indication of its magnitude, measure in radians. The degree of the angle, expressed in radians, is based upon the variety of pi, i beg your pardon expresses the dependence of the one of the circle barisalcity.orge its diameter.

for ease the understanding, i rebarisalcity.orgmend to remember the basic principle the translating degrees in radians. The diameter the the circle covers one arc i beg your pardon is 180 degrees. Thus, the PID will certainly be 180 degrees. Where it is basic to list any type of degree of edge in the radians and also back.

Let"s take into account that the angle of 45 degrees, to express in radians, is equal to (180/45 = 4) π / 4 (four barisalcity.orge four). Therefore, the values ​​found by us are true for the same level angle of the edge expressed in radians:

tangent π / 4 (pi per four) is equal to one sinus π / 4 (pi because that four) levels equals cosine π / 4 degrees and also equals the root of 2 in half

because that the convenience of visual perception, these values ​​are shown in the figure below.

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Note: see alsotable of values of trigonometric functionsof other angles.

Sine, cosine and also tangent angles of 30 degrees (sin cos tg 30) - table of worths |Описание курса| Sine, cosine, tangent of angle 30 and also 60 degrees (sin cos tg 30 and 60)

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