Note. See also a table of values of trigonometric functions for all angles.

Sinus 30, cosine 30, tangent 30 degrees and sinus 60, cosine 60, tangent 60 degrees

How were these values calculated?
You can also go to: 

• tabular values sinus, cosine and tangent 30 degrees, 
• tabular values sinus, cosine and tangent 60 degrees.

This page provides step-by-step explanations for the logic of calculating these values, initially for an angle of thirty degrees, and then sixty. To do this, the construction of an arbitrary rectangular triangle with corresponding angles and calculates the value of the sinus, cosine and tangent of 30 degrees, and in the second part of this article - for a corner of 60 degrees.

The value of trigonometric functions of the sinus, cosine, tangent at α = 30 °

Consider the algebraic triangle ABC. Hey, each of his sides will be equal to a. According to the properties of an equilateral triangle, all its angles are equal, including the angle ∠V = 60 °.

We can calculate the sine, cosine, and tangent values ​​if we find the correlation of the corresponding sides for a 30-degree angle in a rectangular triangle. Since the values ​​of these trigonometric functions depend exclusively on the degree angle of the angle, the correlation we calculate will be the values ​​of the sinus 30, cosine 30, and the tangent 30 degrees.

First we will make additional construction. From the top of A to the BC side we will carry out the median AO.

Median AO in an equilateral triangle is simultaneously bisector and height.

Then the rectangle AOV is a rectangular with an angle ∠AVAO = 30 °. (Angle B is equal to 60 degrees ∠VOA straight and is equal to 90 degrees, so ∠VAO = 180 - 90 - 60 = 30 degrees)

For a given rectangular triangle we calculate the value of the trigonometric functions of its angles. Let's do it first for an angle of 30 degrees.

The size of the hypotenuse is known to us and equal to a. The OB catech is equal to a / 2, since AO is the median of the triangle ABC. We will find a catechist AT.

By Pythagoras's theorem:

АВ²=АО²+ОВ²;

АО²=АВ²-ОВ²

let's substitute in the resulting equation the value of the hypotenuse (we assume that it is equal to a)

АО²=a²- (а/2)²

АО²=3a²/4  

AO=√( 3a²/4 ) =a√3/2

Now we have computed all the sides of the rectangular triangle ABO. Taking into account that AB = a, OB = a / 2, AO = a√3 / 2, we will calculate the obtained values from the relations of the sides of the rectangular triangle. According to the definition of a sinus, a cosine and a tangent:

sin 30 = OB / AB (by definition of sinus - the ratio of the opposing cateta to the hypotenuse)

cos 30 = AO / AB (by cosine definition - the ratio of the adjacent catheter to the hypotenuse)

tg 30 = OB / AO (by the definition of the tangent - the relation of the opposing cateta to the adjoining one)

Reports have:

Given that the triangle ABC is uniform, then BO is AB / 2, and the value of AO is calculated above. As a result, we get tabular values of sin 30, cos 30 and tg 30 degrees

Table values for sin 30, cos 30 and tg 30 degrees:

  

That is:
The tangent of 30 degrees is equal to the root of three to three
A sinus of 30 degrees is equal to one second or 0.5
The cosine is 30 degrees equal to the root of three to two

Taking into account the table of formulas for transformation trigonometric functions,

Since

sin( 90°- 30°) = sin60°, and sin60°=cos30°, then:

Table values for sin 60, cos 60 and tg 60 degrees:

How exactly these functions were calculated is described above. Here are the most important ones:

 

That is:
The tangent of 60 degrees is equal to the root of three
The sinus of 60 degrees is equal to the root of three to two
The cosine is 60 degrees equal to one or two

As can be seen from the calculations given above, in calculating the values of a particular trigonometric function, not specific lengths of the sides are important, but only their ratio, which will always be the same for the same angles, regardless of the size of the triangle.

Sine, cosine and tangent angle of pi divided on 3 (π / 3)

In tasks other than degree angle, angles of angle are often found in radians. The radius of the angle is expressed in terms of the number of pi, which describes the ratio of the length of the circle to its diameter. For the simplicity of remembering the rules for translating the radian in degrees and back, I suggest always remember the following: the diameter of the circle covers an arc of 180 degrees, which is pi radio. Since all values of the angle in radians are indicated by the number of pi, it is sufficient to replace the number of pi by 180 degrees to convert it to a degree degree.

Therefore, take into account that the angle of the py at 3 is equal to 60 degrees. (180/3 = 60)

Where:

• The tangent π / 3 (pi to three) radians is equal to the root of three
• The sine of π / 3 (pi to three) radians is equal to the root of three to two
• The cosine π / 3 (pi to three) is equal to one or two (1/2) or 0.5

To make these values easy to remember visually, they are shown in the picture below.

Sine, cosine, and tangent of the angle of the pi divided on the 6 (π / 6) radians

Proceeding from the above-stated principle of translating the radian in degrees, the angle of the py is equal to 6
180/6 = 30 degrees.

Where:

• The tangent of pi on 6 (π / 6) is equal to the root of three to three
• The sinus pa on 6 (π / 6) equals one second (1/2) or 0.5
• The cosine of pi on 6 (π / 6) is equal to the root of three to two

For a more comfortable visual perception, these values are shown below in the figure

Note. This article shows how the values of sin 60, cos 60 and similar values are calculated. If you want to see a summary of the help information - go to table of values of trigonometric functions.