Reduction to the first quadrant in the trigonometric cycle

When we are working with trigonometry and we come across an angle that is not found in the first quadrant, we can always reduce it in order to find the angle corresponding to this one that is precisely in the 1st quadrant. This is possible thanks to symmetry present in the trigonometric cycle. But we need to pay attention to what happens to the signs of the trigonometric functions in each quadrant.Let's see below some ways to work the quadrant shift in the trigonometric cycle.

Reduction to the First Quadrant

In the following figure, consider the angle x, highlighted in red in the first quadrant. We can find the angles that correspond to x in the other quadrants. The distance of these angles to x is always a multiple of 90°, such that the module of the trigonometric functions of these angles does not change.

Practical method for reduction to the first quadrant
Practical method for reduction to the first quadrant

If the angle we're working with is y and he is in second quadrant, its corresponding in the 1st quadrant will be the angle x such that π - x = y or 180° - x = y.

Example 1:

consider the angle 150°. To reduce it to the 1st quadrant, we will have the following:

180° - x = 150°
x = 30°

Analogously, if the angle y belong to third quadrant, Your correspondent x in the first quadrant will be given by x + π = y or 180° + x = y.

Do not stop now... There's more after the advertising ;)

Example 2:

consider the angle /3, your correspondent will be:

x + π = 3

x = – π
3

x = π3

Finally, if the analyzed angle y belong to fourth quadrant, the angle x corresponding to it in the first quadrant will be given by 2π - x = y or 360° - x = y.

Example 3:

consider the angle 300°, reducing it to the first quadrant, we will have:

360° - x = 300°
x = 60°

Remember that the corresponding angles have similar values ​​of sine, cosine and tangent, and the distinction occurs by the sign. At thefirst quadrant, the values ​​of sine, cosine and tangent are positive. At the second quadrant, O sine is positive, while cosine and tangent are negative.. At thethird quadrant, sine and cosine are negative, while the tangent is positive. At the fourth quadrant, sine and tangent are negative, and cosine is positive.. We can see the distinction between the signs in the following image:

Check the signs of the trigonometric functions according to the quadrant
Check the signs of the trigonometric functions according to the quadrant


By Amanda Gonçalves
Graduated in Mathematics

Would you like to reference this text in a school or academic work? Look:

RIBEIRO, Amanda Gonçalves. "Reduction to the first quadrant in the trigonometric cycle"; Brazil School. Available in: https://brasilescola.uol.com.br/matematica/reducao-ao-primeiro-quadrante-no-ciclo-trigonometrico.htm. Accessed on June 27, 2021.

How to add and subtract fractions?

How to add and subtract fractions?

Fractions represent parts of a whole. From them, addition, subtraction, multiplication and divisi...

read more
Percentage: what it is and how it is calculated (with examples and exercises)

Percentage: what it is and how it is calculated (with examples and exercises)

THE Percentage or Percentage represents a ratio whose denominator equals 100 and indicates a part...

read more
Division: how to do it, which terms and exercises

Division: how to do it, which terms and exercises

Division is a mathematical operation used to discover how to separate a quantity into parts, that...

read more