The Value of Trigonometry Comparison in Different Quadrants

At the beginning of this section, we examine the sine, cosine, tangent and reverse values for domains in units of degrees or radians. In addition, the value of all these comparisons is also learned in each quadrant in Cartesian coordinates.

Let us understand through the following discussion:

Suppose that point A(x, y), length OA = r and angle AOX = α.
Look at the picture above. From right triangle in quadrant I, apply:

sin α = y/r
cos α = x/r
tan α = y/x

Example:
Let A(-12, 5) and ∠XOA = α, Find the value of sin α and tan α!

Answer:
By observing the coordinates of point A(-12, 5), it is very clear that the points are located in the second quadrant, since x = -12, and y = 5.

Geometrically, presented in the picture below:

Since x = -12, and y = 5, using the phytagoras theorem obtained by the oblique side, r = 13. Therefore it is obtained:

sin α = 5/13
tan α = -5/12

Properties of Quadrant Location

If 0 < α < (π / 2), then the value of sine, cosine, and tangent are positive.
If (π / 2) < α < π, then the sine value is positive and the cosine and tangent values are negative.
If π < α < (3π / 2), then the tangent value is positive and the sine and cosine values are negative.
If (3π / 2) < α < 2π, then the cosine value is positive and the sinus and tangent value is negative.