Complementary angles


The word angle , which comes from the Latin word angŭlus , refers to a mathematical figure within the area of geometry that is formed from two lines when they intersect each other on the same surface. The angle then is the region of the plane that is between two rays or sides that have the same origin or vertex. These angles can be measured in specific units and with different measurements as a result and depending on them, they receive a certain classification , and it is important to clarify that the measurement of the angles will always be made in degrees .


What are complementary angles?

Complementary angles are the kind of angle that when added together make a total of 90 degrees . When the angles are complementary they are measured with the right angles .

  • Definition
  • features
  • How to find complementary angles
  • Trigonometric functions of complementary angles
  • Examples


In order to know and understand the meaning of the term complementary angles, we must first know the etymological origin of the words that form it. The word angle is of Greek origin , which derives from the word “ankulos” , which means “crooked” . Then it spread to Latin in the form of ” angulus ” with the meaning of “angle” .


On the other hand, the word complementary is of Latin origin . It is born from the sum of several very well differentiated parts: the prefix “com-” , which means “union” ; the verb “plere” , which is synonymous with “fill” ; “-Ment” , which can be defined as “medium” , and finally, the suffix “-ary” . The latter is used to indicate “relative to” .

That said, it is important to also remember that the angles have different measures and that depending on them, the angles receive their name and classification , in this way, we can say that the complementary angles are those angles that together add up to 90 degrees (90º).


The main characteristics of the complementary angles are the following:

  • Whether they are consecutive or not, they will always add mathematically to 90 degrees .
  • It may be that the angles are not together but if between two angles they manage to make the sum of 90 degrees, then they will be complementary.
  • When two angles add up to 90 degrees , then those angles are considered to complement each other .
  • They are angles that add up to the measure of a right angle .
  • Complementary angles are also composed of two sides and a vertex at the origin each.
  • It is important to know the complementary angles because we can find them in many forms in nature and in many physical phenomena .
  • They can be used in architecture , construction , physiognomy , etc.
  • Two angles do not need to be adjacent to be complementary .

How to find complementary angles

Remembering that complementary angles are those that when added together give 90 degrees or π / 2 rad . Assuming that we have two angles: α = 50⁰ and β = 40⁰, if we add them we will get 90 °, therefore, we say that the angles complement each other. For example, in a right triangle , the sum of the internal angles is equal to 180 °, therefore, we say that in the right triangle the acute angles are considered complementary.

Trigonometric functions of complementary angles

The trigonometric functions are functions that have been established for the purpose of extending the reasons the numbers real and complex . These functions are very important in various areas such as physics , astronomy , cartography and telecommunications . They are generally defined as the quotient between the sides of a right triangle and their relationship to the angles. In the case of complementary angles, let β be the complementary angle of α, where β = 90º – α, the trigonometric ratios of the complementary angle can be obtained as a function of the trigonometric ratios of α.

The trigonometric ratios of the complementary angles are then the following:

  • Sine  of the complementary angle:

sin (90 ° – α) = cos α

  • Cosine of the complementary angle:

cos (90 ° – α) = sin α

  • Tangent  of the complementary angle:

Tan (90 ° – α) = cot α

  • Cosecant  of the complementary angle:

csc (90 ° – α) = sec α

  •  Complementary Angle Secant :

sec (90 ° – α) = csc α

  • Cotangent  of the complementary angle:

cot (90 ° – α) = tan α

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