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Sine, cosine, tangent angle of 105 degrees (sin 105 cos 105 tg 105)
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Sine, cosine and tangent of angle 105 degrees

Синус 105 градусов, косинус 105 градусов, тангенс 105 градусов

How to find the values of trigonometric functions for an angle of 105 degrees

Find the values of the sine, cosine and tangent for an angle of 105 degrees in an analytical way.
At first glance, finding the values of the sine, cosine and tangent for an angle of 105 degrees is a difficult task. However, this is not quite true.
First of all, we must pay attention that for angles whose values exceed 90 degrees, we have formulas reducing to an angle less than 90 degrees.

Therefore, for a start, imagine an angle of 105 degrees as (90 + 15)
Then
sin ( 90 + α ) = cos α
sin 105 = sin( 90 + 15 ) = cos 15

cos ( 90 + α ) = - sin α
cos 105 = cos( 90 + 15 ) = -sin 15

tg ( 90 + α ) = -ctg α
tg 105 = tg( 90 + 15 ) = -ctg 15

Note. Already at this stage, you can see the values in the tables of the sine, cosine and tangent of 105 degrees, which is converted to the values of trigonometric functions of the sine, cosine and tangent of the angle of 15 degrees. But, imagine that we do not know them either.

Therefore, at this stage, we will come to the aid of formulas for transforming the double angle of trigonometric functions..
The fact is that we can imagine an angle of 30 degrees, like a double angle of 15 degrees ( 2 * 15 ).

Then, starting from the identity:
cos 2α = 1 - 2sin2α

We take a double angle as 2 * 15 degrees, then
cos 30 = 1 - 2sin215
sin215 = ( 1 - cos 30 ) / 2

The cosine value for a 30 degree angle is easy to calculate. It is equal to √3/2
sin215 = ( 1 - √3/2 ) / 2
sin 15 = √ (( 1 - √3/2 ) / 2 )

cos 105 = - sin 15 
cos 105 = - √ (( 1 - √3/2 ) / 2 )

by simple algebraic transformations we obtain:

Синус 105 градусов, косинус 105 градусов, тангенс 105 градусов

As can be seen from the example, the values of the trigonometric functions of the sine, cosine and tangent angles of 105 degrees can be obtained by simple trigonometric transformations using trigonometric identities of transformations of double angles and halves of given angles.

See also the full table of values for trigonometric functions (table of sines, cosines and tangents).

Below are also the values of trigonometric functions for an angle of 105 degrees in the form of a decimal fraction with four decimal places.

5π/12
синус 105
 sin 105
косинус 105
cos 105
тангенс 105
tg 105
котангенс 105
ctg 105

Значение

0,9659

-0,2588

-3,7321

-0,2679




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