Trigonometric ratios of 0°, 30°, 45°, 90°, 180° and 270° without calculator

In this lesson, I will teach you how to obtain the trigonometric ratios of 0º (and 360º), 30º, 45º, 60º, 90º, 180º and 270º without using the calculator.

0º and 360º trigonometric reasons

For 0º and 360º we only have one horizontal segment. Therefore, the sine is 0 and the cosine is 1, which is positive because it is to the right of the y-axis:

trigonometric functions of 90 degrees

trigonometric ratios of 90

And the tangent is 0:

Trigonometric ratios of 30º

When the angle is 30º we have a certain value of the sine (green vertical segment) and cosine (blue horizontal segment):

trigonometric functions of 0 degrees

We will deduce how much the values of the sine and cosine are worth and once we have them, we will also have those of the tangent and the cotangent.

If we put aside the triangle formed by the radius of the circumference and the vertical and horizontal segments, we have a right triangle whose smaller angle is 30º and the opposite one is 60º, since the sum of the three angles must be 180º and we know that it has a 90º angle.

We also know that the hypotenuse is worth 1:

trigonometric functions of special angles 30 45 60 and 90

If we double this triangle, we get an equilateral triangle, whose angles are all 60 degrees and therefore, their sides will also be equal, so they will be worth 1:

trigonometric ratios of 180

Then, if we go back to the original right triangle, we see that the opposite leg (vertical side) will be half that in the equilateral triangle, that is, if in the equilateral triangle it was 1 in the original triangle, it will be 1/2:

trigonometric ratios of 30 45 and 60 degree angles

Known the opposite leg and the hypotenuse, we can calculate the value of the adjacent leg with the Pythagorean theorem:

trigonometric ratios of 0 90 180 and 270

trigonometric functions of 30 45 and 60 degrees without calculator

trigonometric ratios of 0 or 30o 45o 60o and 90o

trigonometric ratios of 0o 90o 180o 270 and 360

Therefore, we already know the value of all sides of the right triangle:

trigonometric functions of 30 45 and 60 degree angles exercises

The leg opposite the 30º angle coincides with the sine value of 30º and the leg adjacent to the 30º angle coincides with the cosine value of 30º:

trigonometric ratios of 0o 90o 180 270 and 360

trigonometric ratios for 30 45 and 60 degree angles

Once we know the value of the sine and cosine, we can calculate the value of the tangent of 30º:

trigonometric functions of 0 or 30o 45o 60o 90o

trigonometric ratios quadrants

And from the tangent, we calculate the cotangent of 30º:

trigonometric reasons of a 45 degree angle

trigonometric ratios of 270

As a summary, these are the trigonometric ratios for 30º:

sine cosine and tangent of 30

Trigonometric ratios of 45º

Let us now obtain the value of the trigonometric ratios of 45º. We begin with the sine and cosine, which in principle we do not know their value:

cosine and tangent table up to 360

If we put aside the triangle formed by the radius of the circumference and the vertical and horizontal segments, we have a right triangle whose acute angles measure 45º:

sine cosine and tangent without calculator

At the same time, it is an isosceles triangle, whose base is 1 and whose two equal sides we do not know, which we have called x:

what is the 30 degree sine

Bearing in mind that it is a right triangle and that its two legs measure the same, we will use Pythagoras to calculate how much each of these sides measure:

trigonometric functions of 30 45 and 60 degree angles

trigonometric ratios 30 degrees

trigonometric ratios of 30 45 60

trigonometric functions degrees

30 45 and 60 degree functions

At this point, we rationalize the root and we are left with it:

trigonometric functions of 30 degrees

So we already know the value of both sides:

trigonometric functions of the angles 30 45 and 60

These values therefore correspond to the sine of 45º and the cosine of 45º:

trigonometric angle values

trigonometric ratio values

Once we know the sine of 45º and the cosine of 45º, we can calculate the tangent of 45º:

angles of 30 45 60

value of trigonometric ratios

And knowing the tangent of 45º, we can calculate the cotangent of 45º:

value of trigonometric functions

exact values of trigonometric ratios

These are the trigonometric ratios for 45º:

exact values of trigonometric ratios for the angles 30 45 and 60

Trigonometric ratios of 60º

For the 60º angle, the triangle formed by the radius of the circumference and the vertical and horizontal segments is the same as for the 30º angle, only this time, the sine and cosine values are exchanged:

sine cosine and tangent values table

Therefore, the sine of 60º and the cosine of 60º are valid:

trigonometric functions of 60 degrees

We now calculate the tangent of 60º:

table of cosine and tangent sinuses

razones trigonometricas 30

And the 60-degree cotangent:

cuadrante trigonometrico

trigonometric ratios of 45

The trigonometric ratios for the 60º angle are:

calculate angles without calculator

Trigonometric ratios of 90º

When the angle is 90º we only have a vertical line with a value of 1, positive as it is above the x-axis, which corresponds to the sine. The cosine is therefore 0:

table of trigonometric functions from 0 to 90 degrees

how to get tangent without calculator

The tangent is infinite or does not exist, as it divides by 0:

trigonometrical values of angles

Trigonometric ratios of 180º

For 180 degrees, we only have cosine (horizontal segment) that is worth -1, being to the left of the y-axis. We have no breast, so it’s worth 0

table of trigonometric functions from 0 to 360 degrees

reason trigonometrica de 30

The 180º tangent is 0:

calculate tangent without calculator

Trigonometric ratios of 270º

When the angle is 270º, the sine is -1 and the cosine is 0:

trigonometric values of cosine sine and tangent

how to get the trigonometric ratios

The tangent, as for 90º is also infinite or does not exist, when having to divide by 0:

trigonometric function quadrants

Summary of the main trigonometric ratios of the first quadrant

I leave you with a table with the value of the most important trigonometric ratios of the first quadrant. You should always keep this table handy until you learn it by heart, so that you don’t have to deduct the value of all trigonometric ratios each time:

trigonometric ratios from 0 to 360 degrees

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