Quadrant Identification for Trigonometrical Questions

For question regarding trigonometry, quadrant is one of the key parameter to obtain correct answers.

What is this quadrant about?
A complete cycle (360 degree) is divided into 4 quarters.
They are zones defined for specific trigonometric functions.

The first quarant (0 to 90 degree) gives positive sign for ALL trigonometric functions.
The second quarant (90 to 180 degree) allows only "sine" to have positive number.
For the thrid quarant (180 to 270 degree), "tangent" has positive number only.
Lastly, the fourth quarant (270 to 360 degree), "cosine" gives positive number only.

So, you can see that given a sign of a trigonometrical operation, the specific quadrant can be found or identified.

Example:

sin X = - 0.5 ===> Identifies quadrant as 3rd and 4th.

tan X = 0.2 ===> Identifies the 1st and 3rd quadrant.

This is simple, right?

However, do note the below example.
It causes a mistake that is common!

Example of potential error:

sin 2X = -0.5 ====> which quadrants ?

The answer is not that direct!

Why?

Now the math question is not on "X", but on "2X".

To identify the quadrant, you need to start off from the "2X", working as per normal.
But, after identifying the 2 quadrants, you have to compute the "2X" reference angle.
Using the reference angle, you have to obtain the 2 angles.
After which, you need to divide the angles obtained by 2.
The divided angles is then the final angles lying within the quadrants.

Confused? Never mind. See the numerical solution below.....

Solution:
2X = sin^-1 (0.5) = 30⁰
This is the reference angle used to compute the actual answers.

Final answers are (Quad 3)= 180 + 30 = 210⁰
and (Quad 4) = 360- 30 = 330⁰.

Common mistake is to obtain reference "2X" angle and straight away divide it by 2.
Using this newly found "X", you proceed to identify the angles of the quadrant identified using the "2X". THIS IS INCORRECT!

Do not confuse double angle with single angle.
When the problem is "2X", solve all the way using the "2X" first until reaching the end.
After which, you then divide the angles by 2 to get to the final answers.

Maths is simple if you follow the rules accordingly.
If you mess up double angle with single angle while solving, you just literally mess up the workings.

Maths forces you to follow rules set out. It punishes only if you do not obey orders.

Maths is interesting isn't it? Never expect that maths can police your behaviour while practicing it, right?

:-)