75-Basics of Trigonometry - 03 Important key points

Click here for the previous basics of trigonometry.

Now we will study the next part of trigonometry. 

Today we will study Angles in standard position, positive and negative angles, the definition of trigonometric ratios in the 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 an Anti-Clock-Wise direction through a certain angle keeping point O as a fixed point.

3) The final position of ray OA is the ray OB which is known as a Terminal ray.

4) Here an angle AOB made by these two rays (Ray OA & Ray OB) with fixpoint O as the vertex is known as the directed angle.

Note: 

1) The rotation of the initial ray OA to the terminal ray OB in an anti-clockwise direction from the positive angle.

2) The rotation of the initial ray OA to the terminal ray OB in the clockwise direction from the negative angle.

Standard angle:

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

Trigonometric Ratios in the Cartesian coordinate system:

 
1) Let p (x, y) be any point on the terminal ray OP (also called "arm").

2) Let OP = r.

3) By the theorem of Pythagoras, we have 

O A ² + A P ² = O P ² 

 so, x ² + y ² = r ² 

Now we will define all the trigonometric ratios using x, y, and r as shown below.

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 the 2nd quadrant, "x" is negative, and "y", and "r" is positive. So only the ratios sin and csc contain "y" and "r" so they are positive and the rest are 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 the rest all are negative.

4) In the 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 the 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 the following statements:

1) All Silver Tea Cups                                 (All, sin, tan, cos)     

2) Add Sugar To Coffee                              (All, sin, tan, cos)      

3) All Scholars Take Computer-Science   (All, sin, tan, cos)     

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

The next part of this topic will be published in the next Blog.

Click here for the next basics of trigonometry.

Anil Satpute