Sine, cosine and tangent of angle 105 degrees
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 - 2sin
215
sin
215 = ( 1 - cos 30 ) / 2
The cosine value for a 30 degree angle is easy to calculate. It is equal to √3/2
sin
215 = ( 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:
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
|