77-Basics of Trigonometry - 05 Important key points

Click here for the previous basics of trigonometry.

Basics of trigonometry

Today we will study trigonometric ratios of 90°, 180°, and 270°.

5) Trigonometric ratios of an angle O = 90°

Here angle AOP is 90°.

According to the diagram, OP = r, and the coordinates of point P are (0, r). So x-coordinate of point P is 0 and the y-coordinate of point P is r.

So, all the trigonometric ratios of O = 90° with

 x = 0, y = r, r = r.

a) sin 90° = y/r     sin 90°  = r/r        sin 90°  = 1	b) cos 90° = x/r      cos 90° = 0/r      cos 90° = 0	c) tan 90° = y/x     tan 90°  = r/0       tan 90°  = ∞
d) csc 90° = r/y     csc 90° =  r/r     csc 90° =  1	e) sec 90° = r/x     sec 90° = r/0          sec 90° = ∞	f) cot 90° = x/y     cot 90° = 0/r        cot 90° = 0

6) Trigonometric ratios of an angle O = 180°

Here angle AOP is 180°.

According to the diagram, OP = r, and the coordinates of point P are (-r, 0). So the x-coordinate of point P is -r and the y-coordinate of point P are 0.

So, all the trigonometric ratios of O = 180° with

 x = -r, y = 0, r = r.

a) sin 180° = y/r     sin 180°  = 0/r        sin 180°  = 0	b) cos 180° = x/r      cos 180° = -r/r      cos 180° = -1	c) tan 180° = y/x      tan 180° = 0/-r        tan 180° = 0
d) csc 180° = r/y     csc 180° =  r/0     csc 180° =  ∞	e) sec 180° = r/x     sec 180° = r/-r              sec 180° = -1	f) cot 180° = x/y     cot 180° = -r/0      cot 180° = -∞

7) Trigonometric ratios of an angle O = 270°

Here angle AOP is 270°.

According to the diagram, OP = r, and the coordinates of point P are (0, -r). So x-coordinate of point P is 0 and the y-coordinate of point P is -r.

So, all the trigonometric ratios of O = 270° with

 x = 0, y = -r, r = r.

a) sin 270° = y/r     sin 270° = -r/r        sin 270° = -1	b) cos 270° = x/r      cos 270° = 0/r      cos 270° = 0	c) tan 270° = y/x     tan 270° = -r/0       tan 270° = ∞
d) csc 270° = r/y     csc 270° =  r/-r     csc 270° =  -1	e) sec 270° = r/x     sec 270° = r/0     sec 270° = ∞	f) cot 270° = x/y    cot 270° = 0/-r       cot 270° = 0

Actually, it is very easy to understand the concept once we found out the values of x, y, and r for the point P on the terminal ray.

In the next Blog, we will study all the trigonometric ratios for the following groups.

Group-01: 120° (90° + 30°), 210° (180° + 30°), 300° (270° + 30°).

Group-02: 135° (90° + 45°), 225° (180° + 45°), 315° (270° + 45°).

Group-03: 150° (90° + 60°), 240° (180° + 60°), 330° (270° + 60°).

Click here for the next basics of trigonometry.

Anil Satpute

76-Basics of Trigonometry - 04 Important key points

Click here for the previous basics of trigonometry.

Now we will study the next part of trigonometry. 

Today we will study trigonometric ratios of 0°, 30°, 45°, and 60°.

1) Trigonometric ratios of an angle q =  0°

Here angle AOP is 0°.

According to the diagram, OP = r, and the coordinates of point P are (r, 0). So the x-coordinate of point P is r and the y-coordinate of point P is 0.

So, all the trigonometric ratios of q = 0° with

 x = r,

 y = 0,

 r  = r.

a) sin 0° = y/r     sin 0° = 0/r        sin 0° = 0	b) cos 0° = x/r     cos 0° = r/r        cos 0° = 1	c) tan 0° = y/x     tan 0° = 0/r        tan 0° = 0
d) csc 0° = r/y     csc 0° = r/0          csc 0° = ∞	e) sec 0° = r/x     sec 0° = r/r        sec 0° = 1	f) cot 0° = x/y     cot 0° = r/0        cot 0° = ∞

2) Trigonometric ratios of an angle q = 30°

Here angle AOP is 30°.

We know that the side opposite to 30° is half the hypotenuse and the side opposite to 60° is √3/ 2 times the hypotenuse.

So if hypotenuse OP = r, then  AP = r/2 and OA = (√3 r)/ 2.
Here the coordinates of point P will be P ((√3 r)/ 2, r/2).

So, all the trigonometric ratios of q = 30° with

 x = (√3 r)/ 2,

 y = r/2,

 r = r.

a) sin 30° = y/r     sin 30° = (r/2)/r        sin 30° = 1/2	b) cos 30° = x/r     cos 30° = [(√3 r)/2]/r            cos 30° = √3/2	c) tan 30° = y/x     tan 30° = (r/2)/(√3 r)/2        tan 30° = 1/√3
d) csc 30° = r/y     csc 30° =  r/(r/2)          csc 30° = 2	e) sec 30° = r/x     sec 30° = r/[(√3 r)/2]        sec 30° = 2/√3	f) cot 30° = x/y     cot 30° = [(√3 r)/ 2]/(r/2)        cot 30° = √3

3) Trigonometric ratios of an angle q = 45°

Here angle AOP is 45°.

We know that the side opposite 45° is 1/√2 times the hypotenuse.

So if hypotenuse OP = r, then  AP = r/√2 and OA = r/√2.
Here the coordinates of point P will be P (r/√2, r/√2).

So, all the trigonometric ratios of q = 45° with

 x = r/√2,

 y = r/√2,

 r = r.

a) sin 45° = y/r      sin 45°  = [r/√2]/r         sin 45°  = 1/√2	b) cos 45° = x/r      cos 45° = [r/√2]/r      cos 45° = 1/√2	c) tan 45° = y/x      tan 45°  = [r/√2]/ [r/√2]        tan 45°  = 1
d) csc 45° = r/y     csc 45° =  r/[r/√2]     csc 45° = √2	e) sec 45° = r/x     sec 45° = r/[r/√2]        sec 45° = √2	f) cot 45° = x/y     cot 45° = [r/√2]/ [r/√2]        cot 45° = 1

4) Trigonometric ratios of an angle q = 60°

Here angle AOP is 60°.

We know that the side opposite 60° is √3/2 times the hypotenuse and the side opposite 30° is half times the hypotenuse.

So if hypotenuse OP = r, then  AP = (√3 r)/2 and OA = r/2.
Here the coordinates of point P will be P (r/2, (√3 r)/2).

So, all the trigonometric ratios of q = 60° with

 x = r/2,

 y = (√3 r)/2,

 r = r.

a) sin 60° = y/r     sin 60° = [(√3 r)/2]/r        sin 60° = √3/2	b) cos 60° = x/r      cos 60° = [r/2]/r      cos 60° = 1/2	c) tan 60° = y/x     tan 60° = [(√3 r)/2]/[r/2]       tan 60° = √3
d) csc 60° = r/y     csc 60° =  r/[(√3 r)/2]     csc 60° = 2/√3	e) sec 60° = r/x     sec 60° = r/[r/2]        sec 60° = 2	f) cot 60° = x/y     cot 60° = [r/2]/[(√3 r)/2]        cot 60° = 1/√3

In the next blog, we will study the trigonometric ratios of 90°, and 180° and the tabulated form of all the trigonometric ratios.

Click here for the next basics of trigonometry.

Anil Satpute

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

74-Basics of Trigonometry - 02 Important key points

Click here for the previous basics of trigonometry.

Trigonometry is an important topic for your higher studies. We must study this topic with all basics. Actually, this topic is very simple. If we concentrate to learn this topic from the beginning, we will definitely understand the basics of this topic and it will help us to remember the formulas (formulae) very easily. At the end of all the Blogs related to “Basics of trigonometry”, we will list all the formulas (formulae) in a separate sheet so that it will be helpful for you to remember all these formulas (formulae).

1) Define all trigonometric ratios for an angle C of the above triangle ABC.  

Answer: 
1) sin C  = AB/AC
2) cos C = BC/AC
3) tan C  = AB/BC
4) csc C = AC/AB
5) sec C  = AC/BC
6) cot C  = BC/AB  

2) Observe the adjacent figure and write down all the trigonometric ratios.
Answer:
1) sin P   = RQ/PQ          1) sin Q   = PR/PQ
2) cos P  = PR/PQ           2) cos Q  = RQ/PQ 
3) tan P   = RQ/PR           3) tan Q  = PR/RQ
4) csc P  = PQ/RQ           4) csc Q  = PQ/PR
5) sec P  = PQ/PR            5) sec Q = PQ/RQ
6) cot P   = PR/RQ           6) cot Q  = RQ/PR  
In the above example, just observe 
case 1) sin P and cos Q
case 2) cos P and sin Q

Here, the basic point to understand is that the comparison of two angles is given with different trigonometric ratios. First, observe these two angles. angle P and angle Q. In a right-angled triangle, QPR, Angle R = 90, so the remaining two angles P and Q are complementary angles. 

Here all of us will remember the following trigonometric relations of complementary angles of a right-angled triangle.

1) sin A  = cos (90-A)
2) cos A = sin (90-A)
3) sec A = csc (90-A)
4) csc A = sec (90-A)
5) tan A  = cot (90-A)
6) cot A  = tan (90-A)

A) An easy way to remember the above relations:

The relation between the trigonometric ratios of co-terminal angles. (See the following figure)

Trigonometric Ratios of an angle A	Trigonometric Ratios of Complimentary Angle of A. (90°-A)	Ratio (In the form of the length of the sides of a triangle)
sin A	cos (90°-A)	BC/AC
cos A	sin (90°-A)	AB/AC
csc A	sec (90°-A)	AC/BC
sec A	csc (90°-A)	AC/AB
tan A	cot (90°-A)	BC/AB
cot A	tan (90°-A)	AB/BC

B) Relation between the trigonometric ratios:

Observe the following trigonometric ratios from the figure given below:

1) sin C  = AB/AC

2) cos C = BC/AC

3) tan C  = AB/BC

4) csc C = AC/AB

5) sec C  = AC/BC

6) cot C  = BC/AB

If we divide equation 1 by equation 2 we get equation 3. i.e. 

                            

In the same way, we can prove that "cos C / sin C  =  cot C"

In the Next Blog, we will study the trigonometric ratios of some standard angles and trigonometric identities.

Click here for the next basics of trigonometry.

Anil Satpute