Tuesday, December 24, 2013

Basics of Trigonometry - 03

Dear Students,
Now we will study the next part of trigonometry. 

Today we will study Angles in standard position, positive and negative angles, definition of trigonometric ratios in Cartesian coordinate system and Signs of trigonometric ratios in different quadrants.

Angles in standard position:

1) Ray OA is an Initial Ray.
2) Rotate ray OA in Anti-Clock-Wise direction through certain angle keeping point O as fixed point.
3) Final position of ray OA is the ray OB which is known as Terminal ray.
4) Here an angle AOB made by these two rays (Ray OA & Ray OB) with fix point O as the vertex is known as the directed angle.

1) The rotation of initial ray OA to the terminal ray OB in anti-clock-wise direction form the positive angle.
2) The rotation of initial ray OA to the terminal ray OB in clock-wise direction form the negative angle.

Standard angle:
In Cartesian coordinate system, the amount of rotation of initial ray OA to the terminal ray OB through an origin is known as the standard angle.

Trigonometric Ratios in Cartesian coordinate system:

1) Let p (x, y) be any point on the terminal ray OP (also called as "arm").
2) Let OP = r.
3) By theorem of Pythagoras, we have 

O A 2 + A P 2 = O P 
 so, x 2 + y 2 = r 2 
Now we will define all the trigonometric ratios using x, y and r as shown bellow.

1) sin q = y / r 
2) cos q = x / r 
3) tan q = y / x 
4) csc q = r / y 
5) sec q = r / x 
6) cot q = x / y 

Signs of trigonometric ratios in different quadrants:                

1) All the ratios in the 1st quadrant are positive.
2) In 2nd quadrant, "x" is negative and "y", "r" are positive. So only the ratios sin and csc contain "y" and "r" so they are positive and the rest all negative. 
3) In 3 rd quadrant, both "x" and "y" are negative and "r" is positive. Here, only tan and cot contain "x" and "y". As both "x" and "y" are negative, their division will be positive, so tan and cot will be positive and rest all are negative.
4) In 4th quadrant, "x" and "r" are positive and "y" is negative. Here only cos and sec contain "x" and "r" , so cos and sec are positive and rest all are negative.

In short,

      1st  quadrant                    all ratios are positive
      2nd quadrant               "sin" and "csc" are positive
      3rd  quadrant               "tan" and "cot" are positive
      4th  quadrant               "cos" and "sec" are positive 

To remember this case we can prepare following statements:

1) All Silver Tea Cups                                 (All, sin, tan, cos)     
2) Add Sugar TCoffee                              (All, sin, tan, cos)      
3) All Scholars Take Computer-Science   (All, sin, tan, cos)     

All the students need to remember above statements so that we will definitely remember the signs of the trigonometric rations in different quadrants.

Next part of this topic will be published in next Blog.

Anil Satpute