Trigonometry (Class X to XII)

What is trigonometry? 

The word"trigonometry" is a combination of Greek word 'tri', 'gon' and 'matron' where the words means three, arm and measure respectively. Trigonometry is the branch of mathematics that deals the measurement of the sides and angles of a triangle and the relationship between them. About 2000 years ago, the famous Greek astronomer Hipparchus named the branch of mathematics trigonometry. This subject has been discussed in the ancient Hindu book Suryasiddhanta. 

Uses of trigonometry :

In fact, in ancient times the method of measuring the height of a large hill or the width of a large river was not known or it was a very difficult task.

 But now with the help of trigonometry these tasks can be done very easily and effortlessly. Currently trigonometry is a large practical and useful branch of mathematics. It is also diverse in our daily life

Widespread use is undeniable. 

 

Trigonometric angles :

There a question occurs of positive and negative angles in the case of trigonometric angles. (Geometric angles are all positive) 

In this case , if the rotating ray rotates anticlockwise, it produces positive angles.

In the above figure, the angle AOB is positive, because here the ray rotates counterclockwise. 



If the ray rotates clockwise, it produces negative angles. In the above mentioned figure the angle MOG is negative, because OM rotates clockwise. 

In the case of trigonometric angles, a complete rotation of the ray results in an angle of 360 . Thus if a beam rotates at a 60° angle again after a complete rotation on the opposite side of the clock face, then the net generated angle is =360°+60°=420°. The same is true for other rotations of the ray. It is possible to obtain positive or negative angles greater than 360.

Trigonometric Ratios :

  

Trigonometric ratios are the ratio of two sides for the corresponding angle. 

In the figure bellow, AB is opposite to the angle ABC = θ, AB is adjacent , AC is hypotenuse of the right angled triangle 📐ABC. 

Note: If we take the angle BAC= Φ , then AB and BC will be the adjacent and opposite respectively. 

The six trigonometric ratios are sine, cosine, tangent, cosecant, secant, cotangent. The first three are the basic trigonometric ratio.  Depending on the angle θ in triangle ABC, 

 sine = sinθ=opposite/hypotenuse=AB/AC

 cosine=cosθ=adjacent/hypotenuse=BC/AC

tangent=tanθ=opposite/adjacent=AB/BC

cosecant=cosecθ=hypotenuse/opposite=

AC/AB

secant=secθ=hypotenuse/adjacent=AC/BC

 

cotangent=cotθ=adjacent/opposite=BC/AB

Important notes:

1) sinθ ≠ sin×θ ,

2) (sinθ)² ≠ sin(θ²) , 

3) (sinθ)²=sin²θ , 

4) value of sinθ and cosθ can never greater than 1 .(-1 ≤sinθ≤1 and -1 ≤cosθ≤1) 

Because ,

                  sinθ=opposite/hypotenuse

                 

                  cosθ=adjacent/hypotenuse

Length of hypotenuse is greater than or equals to the opposite or adjacent. So the maximum value of sinθ , cosθ is 1.

5)  tan, cosec, sec, cot can take any values. 

(-∞ ≤tanθ≤∞) 

Relations between the trigonometric ratios :



To know the relations between the trigonometric ratios from the above figure, we see;

sin θ = opposite/hypotenuse = AB/AC and

cosec θ = hypotenuse/opposite =AC/AB

It is clear that one is the reciprocal of the other.

So, sin θ = 1/cosec θ and

cosec θ = 1/sin θ ………. (a)

Again, cos θ = adjacent/hypotenuse = BC/AC and

sec θ = hypotenuse/ adjacent=AC/BC

One is reciprocal of the other.

That is, cos θ = 1/sec θ and sec θ = 1/cos θ ………. (b)

So, tan θ = opposite/adjacent = AB/BC and cot θ = opposite/adjacent = BC/AB

tan θ = 1/cot θ and cot θ = 1/tan θ ………. (c)

Moreover, sin θ/cos θ = (AB/AC) ÷ (BC/AC) = (AB/AC) × (AC/BC) = AB/BC = tan θ

Therefore, sin θ/cos θ = tan θ ………. (d)

and cos θ/sin θ = (BC/AC) ÷ (AB/AC) = (BC/AC) × (AC/AB) = BC/AB= cot θ

Therefore, cos θ/sin θ = cot θ ………. (e)

                                                                                            

 From pythagoras's theorem we can write from 🔺ABC , 

                                AC² = AB² + BC²............. (A) 

           Dividing both sides by AC², 

                         =>     1=(AB/AC) ²+(BC/AC) ²

                         =>      sin² θ + cos² θ = 1...............(1) 

          Dividing both sides by BC², 

                         

                         =>     (AC/BC) ²=(AB/BC) ²+1

                         =>     sec² θ = tan² θ + 1.............(2) 

              

             Dividing both sides by AB², 

        

                          =>     (AC/AB) ²=1 + (BC/AB) ²

           

                        =>    cosec²θ=1 + cot² θ .............(3) 

                         (i) 1 - cos² θ = sin² θ 

                         (ii) 1 - sin² θ = cos² θ


                         (iii) sec² θ - 1 = tan² θ 

                         (iv) sec² θ - tan² θ = 1


                         (v) cosec² θ - 1 = cot² θ 

                          (vi) cosec² θ - cot² θ = 1

Values of trigonometric functions :



The values of 0°, 30° ,45°, 60°, 90° are widely used values. 

 

Complimentary trigonometric angles :

 

sin (90°-θ )= cosθ 

cos (90°-θ )= sinθ 

sec (90°-θ  ) = cosecθ 

Cosec (90°-θ )= Secθ 

Fam (90°-θ ) =cotθ 

cot (90°-θ ) = tanθ 

Trigonometric ratios of negative angles :

(i) sin (- θ) = - sin θ  

 (ii) tan (- θ) = - tan θ

(iii) cos (- θ) = cos θ

(iv) cosec (- θ ) = – cosec θ

v) sec(- θ)  = sec θ

 

(vi) cot  (- θ ) = - cot θ

~ cos θ  and sec θ are even trigonometric functions

and remainings are odd trigonometric functions. 

Sign of trigonometric ratios :

1st quadrant : 

All trigonometric ratios are positive. 

2nd quadrant

Only values of sin and cosec are postive. 

3rd quadrant

Only values of tan and cot are positive. 

4th quadrant

Only values of cos and sec are positive. 

Compound angles:

Trigonometric ratios of compound angles :

(i) sin(A + B) = sinA cosB + cosA sinB 

(ii) sin(A - B) = sinA cosB- cosA sinB. 

(iii) cos(A + B) = cosA cosB -sinA sinB.

(iv) cos(A - B) = cosA cosB + sinA sinB.

(v) tan(A +B) = tanA + tanB /1- tanAtanB

(vi) tan(A-B) = tanA - tanB /1 + tanAtanB

(vi)  cot (A+ B) = cot A cot B-1/cot B + cotA

(viii) cot (A – B) = cotA cotB + 1 /cot B - cotA

(ix) tan(A+B+C) =

    ( tanA+tanB+tanC - tanA tanBtanC) /(1-tanAtanB-tamBtanC-tanCtanA) 

(x) 

sin(A+ B) sin (A- B)=sin²A -sin²B =cos²B -cos²A

(xi) 

cos (A + B) cos (A - B)= cos²A - sin²B=cos²B-sin²A

Transformation of sums and products of trigonometric ratios :

  

Sum to product :

(i)  sin C+ sin D = 2 sin (C + D)/2 cos (C - D)/2     

(ii)  sin C - sin D = 2 cos (C + D)/2 sin (C - D)/2 

(iii)  cosC  + cos D= 2 cos (C+ D)/2 cos (C- D)/2    

(iv ) cos C - cos D = 2 sin (C + D)/2 sin (D - C)/2    

Product to sum:

(i) 2 sin C cos D= sin (C + D) + sin (C- D)

(ii) 2 cos Csin D= sin (C+ D) - sin (C- D) 

(iii) 2 cos C cos D= cos (C + D) + cos (C - D)

(iv) 2 sin C sin D = cos (C- D) - cos (C + D) 

Trigonometric ratios of multiple angles :

1) sin2A = 2sinAcosA=2tanA/(1+tan²A) 

2) cos 2A = cos²A - sin²A = 2cos²A-1 = 1-2sin²A

3) tan 2A =2tanA/(1- tan²A) 

4) tan²A = (1-cos 2A)/(1+ cos 2A) 

5) cos2A=(1-tan²A) /(1+tan²A) 

6) sin3A = 3 sin A - 4sin³A

7) cos 3A = 4cos³A - 3 cos A

8)tan 3A = (3 tanA - tan³ A) /(1- 3 tan²A) 

Trigonometric ratios of submultiple angles :

1) sinA = 2sin(A/2)cos(A/2)=2tan(A/2)/(1+tan²(A/2) ) 

2) cos A = cos²(A/2)- sin²(A/2) = 2cos²(A/2)-1 = 1-2sin²(A/2) 

3) tan A =2tan(A/2)/(1- tan²(A/2)) 

4) tan²(A/2)= (1-cos A)/(1+ cos A) 

5) cosA=(1-tan²(A/2)) /(1+tan²(A/2)) 

6) sinA = 3 sin (A/3) - 4sin³(A/3) 

7) cos A = 4cos³(A/3 ) - 3 cos (A/3) 

8)tan 3A = (3 tan(A/3)- tan³ (A/3)) /(1- 3 (tan²A/3)) 

9)(1+sinA) =(sin(A/2)+cos(A/2))²

10)(1-sinA) =(sin(A/2)-cos(A/2))²