I have seen students cramming the signs of Trigonometry Functions ( sinϴ, cosϴ, tanϴ, cosecϴ, secϴ, cotϴ) in four Quadrants. Very few students know the concept behind it. But should always focus on conceptual knowledge hence below mentioned is the concept behind signs.

Above shown is a Unit Circle means a simple circle with Radius=1. 1 can have any measuring unit (mm,cm,m etc). AC is radius and BC is a perpendicular drawn from point C on a simple line passing from the center of the circle. Let us call this line as x axis and a line perpendicular to x axis called as y axis.

We see a triangle is formed where AC is the Hypotenuse, BC is the Perpendicular and AB is the Base. Let us call the angle between AC and AB as ϴ.

Now By Definition,

sinϴ    = Perpendicular/Hypotenuse

= BC/AC

As this is a Unit Circle which means AC=1 thus

sinϴ    = BC

Now If we understand what is BC it is actually Projection of AC on y axis which implies Sinϴ is Projection of AC on y axis

Hey we just understood what is Sinϴ.. that’s so easy ..

Similarly

cosϴ   =Base/Hypotenuse

=AB/AC

As this is a Unit Circle which means AC=1 thus

cosϴ   =AB

And AB is nothing but Projection of AC on x axis which implies Cosϴ is Projection of AC on x axis.

Signs of Sinϴ,Cosϴ in the 4 Quadrants

As we understood above Sinϴ is Projection of AC on y axis and Cosϴ is Projection of AC on x axis

1. In Quadrant 1 (Filled in Green) both y and x are positive and thus Sinϴ, Cosϴ both have positive values.
2. In Quadrant 2 (Filled in Yellow) y is positive thus Sinϴ is positive, but x is negative so Cosϴ has negative value.
3. In Quadrant 3 (Filled in Orange) both y and x are negative and thus Sinϴ, Cosϴ both have negative values.
4. In Quadrant 4 (Filled in Blue) y is negative and thus Sinϴ is negative and x is positive so Cosϴ has positive value.

Now we can easily calculate signs of tanϴ, cotϴ, cosecϴ, secϴ..