Relation between angle ratios of different quadrants

Next, I’m going to show you the relation between the trigonometric ratios of angles of different quadrants, which we use to calculate the ratios of angles that are related to an angle of the first quadrant we already know.

Not only will I limit myself to quoting the formulas that relate them, but I will also explain to you where each of them comes from so that you can understand them better.

Supplementary and supplementary angles

Before you begin to analyze the relationships between the ratios of related angles, you should know two concepts: complementary angles and supplementary angles.

What are complementary angles?

The complementary angles are those whose sum is equal to 90º.

For example:

40º and 50º are complementary angles since their sum is equal to 90º.

What are supplementary angles?

The supplementary angles are those whose sum is equal to 180º.

For example:

45º and 135º are seplementary angles since their sum is equal to 180º.

Relation between complementary angle ratios

When two angles add up to 90º, the cosine lengths of α and the sine length of 90º-α are equal. Note that 90º-α is equal to another angle α, but measured from the y-axis:

relations between the ratios of certain angles

And therefore:

relationship between angles

The sine lengths of α and the cosine of 90º-α are also equal:

angle ratio

So, as far as I’m concerned:

relationship between angles examples

The tangent between these two angles is also related, although it cannot be seen graphically:

relationships between angles

For example:

30º and 60º are complementary angles since 90-60=30, therefore:

angle ratios

Relation between supplementary angle ratios

When the angles are supplementary the sine length of α is equal to the sine length of 180º-α, since α is at the same distance of 0º as 180º-α is at 180º.

On the other hand, the cosines have the same value, but changed their sign:

relationship between angles

Therefore:

trigonometric ratios quadrants

relationships between trigonometric ratios of some angles

And the tangent, although not seen graphically, is thus related between these two angles:

relationships between trigonometric ratios of an angle

For example:

The 30º and 150º angles have this relationship. 30º is at the same distance from 0º, that 150º is at 180º and then:

cuadrantes y angles

Relation between ratios of angles that differ by 180º

When the angles differ by 180º the length of the sine of α is equal to the length of the sine of 180º+α, but of opposite sign, since α, is at the same distance of 0º as 180º+α is of 180º and besides the sine of 180º+α is negative since it falls below the x axis.

On the other hand, the cosines have the same value, but changed their sign:

signs and values in different quadrantsTherefore:

angle relationships

relacion angles

And the tangent, although not seen graphically, is thus related between these two angles:

angulo -564 cuadrante

For example:

The 30º and 210º angles are in this relationship. 30º is at the same distance from 0º, that 210º is at 180º and then:

relationship between trigonometric ratios

Relation between ratios of opposite angles

When the angles are opposite, i.e. α and -α, the value of your cosine is the same and that of your sinuses has an opposite sign:

related angle

Therefore:

trigonometric functions in quadrants

angle ratios

And for the tangent:

angulos relacionadosFor example:

30º and -30º are opposite angles. So:

complementary angle ratios

Relation between angle ratios differing by 90º

For angles that differ by 90º, the cosine of α has the same length as the sine of 90º+α, both positive signs, and the sine of α has the same length as the cosine of 90º+α, but the latter is negative because it is to the left of the y-axis:

relationships between trigonometric ratios

Therefore:

angles by quadrants

angle relationships

And the tangent of angles that differ by 90º is related in this way:

relationship between angles

For example:

30º and 120º are differentiated by 90º, therefore it will be fulfilled that:

pasar angulos al primer cuadrante

As you can see, by knowing the value of the ratios in the first quadrant, you may be able to calculate angle ratios in the rest of the quadrants with these relationships.

Applications of ratio ratios between different angles.

What good is knowing these relationships to you? Well, let’s look at it with a few examples.

The calculator only gives you an angle of the possible

As we have seen before with this performance:

trigonometric ratios in quadrants

Two different angles can have the same sine value. As a matter of fact:

relationship between trigonometric ratios

On the other hand, if what they give you is the value of the breast, you have to keep in mind that the calculator will only give you the result from one of the two possible angles.

For example, they give you this value:

trigonometria angulos complementarios

And they ask you to calculate the value of α.

The calculator, with the arc sine (inverse sine) function, will only give you one of the two possible values:

trigonometric functions complementary angles

The other, you must calculate it yourself, knowing that the angle 180º-α has the same breast:

cuadrante angles

And if we solve the sine of this last angle, we can see that the value of the sine is the mime:

quadrants of the angles

If you are given the cosine value to calculate the angle, the same thing happens: there will be two angles that have that cosine value:

angles in quadrants

For example, they give you this cosine value:

cuadrante de angles

And they ask you to calculate the value of α.

Again, the calculator will only give you one of the possible values:

trigonometric reasons supplementary angles

The other possible value is its opposite angle -72.54º or else:

related angles

As we see on the graph of the circumference.

If we check it we see that the 3 angles have the same cosine value:

quadrant of angles

We can calculate trigonometric ratios of related angles

Another application is that if we know for example the sine value of 30, we are able to calculate the trigonometric ratios of its related angles, that is to say:

  • Angles that are around 0º (0º+30º, 0-30º): 30º and -30º
  • Angles that are around 90º (90º+30º, 90-30º): 120º and 60º
  • Angles that are around 180º (180º+30º, 180-30º): 210º and 150º
  • Angles that are around 360° (360°-30°): 330°

trigonometric reasons of supplementary angles

You are now able to obtain trigonometric ratios from many angles without using the calculator.

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