Basic Trigonometry

Angle Measurement :
1. 1 radian angle (1ͨ) = 90 degrees (90°)
2. 1° = 60 minutes (60')
3. 1' = 60 seconds (60")
4. 1 right angle = 100 grades =100ᵍ
5. 1ᵍ = 100'
6. 180° = 200ᵍ = π radians
1 radians = 57° (approx.)
Trigonometric ratios:
1. (i) tanθ = sinθ/cosθ, Cotθ = cosθ/sinθ
(ii) sinθ.cosecθ = 1
(iii) cosθ.secθ= 1
(iv) tanθ.cotθ= 1
2. (i) sin²θ + cos²θ = 1
(ii) sec²θ – tan²θ = 1
(iii) cosec²θ – cot²θ  = 1
3. |sinθ| ≤ 1, |cosθ| ≤ 1, |cosecθ| ≥ 1, |secθ| ≥ 1 for all θ ϵ R.
Quadrant Rule :
90° = All trigonometric ratios are +ve
180° =  Sin and Cosec are +ve and others are –ve
270° =  Tan and cot are +ve and others are –ve
360° =  Cos and Sec are +ve and others are -ve

Trogonometric ratio of some Standard Angles :


1) Sin(90-θ) = cosθ
Cos(90-θ)= sinθ
Tan(90-θ)= cotθ
2) Sin(90+θ) = cosθ
Cos(90+θ)= -sinθ
Tan(90+θ)= -cotθ
3) For  n ϵ I
sin(n.360°+θ)=sinθ
cosec(n.360°+θ)=cosecθ
cos(n.360°+θ)=cosθ
sec(n.360°+θ)=secθ
tan(n.360°+θ)=tanθ
cot(n.360°+θ)=cotθ

Formulae for 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).
vii. cot(A+B)= (cotA.cotB – 1)/(cotB + cotA).
viii. cot(A-B) = (cotA.cotB + 1)/(cotB – cotA).
ix. sin(A+B).sin(A-B) = sin²A – sin²B.
x.  cos(A+B) = cos²A-cos²B.
Multiple  and  submultiple  angles :-
i. sin2A = 2sinA.cosA = 2tanA/(1 + tan²A).
ii. cos2A = cos²A-sin²A = 1-2sin²A = 2cos²A-1 = (1 – tan²A)/(1 + tan²A).
iii. tan2A = 2tanA/(1 – tan²A .
iv. sin3A = 3sinA – 4sin³A.
v. cos3A = 4cos³A – 3cosA.
vi. tan3A = (3tanA – tan3A)/(1 – 3tan²A).
vii. cot3A = (cot³A – 3cotA)/(3cot²A – 1).
Changing  product  into  sum  or  difference  and  vice –versa :
1. 2sinA.cosB = sin(A+B) + sin(A-B).
2. 2cosA.sinB = sin(A+B) – sin(A-B).
3. 2cosA.cosB = cos(A+B) + cos(A-B).
4. 2sinA.cosB = cos(A-B) – cos(A+B).
5. sinC +sinD = 2sin{(C+D)/2}.cos{(C–D)/2}.
6. sinC – sinD = 2cos{(C+D)/2}.sin{(C–D)/2}.
7. cosC + cosD = {(C+D)/2}.sin{(C–D)/2}.
8. cosC – cosD = 2sin{(C+D)/2}.sin{(D–C)/2}.
Even function :-
A function of the form f(-x) = f(x) is known as even function. Eg.
sec(-θ) =secθ     cos(-θ)= cosθ
f(x) =x2, f(x) = |x| etc.
Note :-
i. Derivative of the even function is odd function.
Even function : f(-x) = f(x)
Differentiation: f '(-x) = -f '(x)  (Odd function)
ii. If x is replaced by –x in an even function
there is no change in the equation of the 
curve. i.e. f(x) = x²
putting x=-x , f(-x) = (-x) ² = x²
iii. Portion of the curve lying on the either side of x-axis are equal. i.e. even function is symmetric about y-axis.
Odd function : A function of the form f(-x)=-f(x) is known as odd function. e.g.: sin(-A)=-sinA, tan(-A)=-tanA, f(x) = x5, f(x) = xcosX etc.
Note :
i. Derivative of the odd function is an even function. i.e.
f(-x) = -f(x)
Differentiating:
f'(-x)(-1)=-f'(x)
f'(-x)=f'(x)
i.e.f'(x) is an even function.
ii. Constant function is both even and odd function.
Periodic function : The function of the form f(x+p)=f(x), where p is least positive number is called periodic function.
For sinx, cosx, p =2π
For tanx, cotx, p = π
i.e. sin(2π + θ)= sinθ, cos(2π +θ) =cosθ, tan(π + θ) = tanθ
The periodic function of cos(ax+b) is 2π/a.
Tips:
- π is an irrational number. π ≈ 22/7 =3.1414…….
- In the result θ = l/r; θ is always in radians wheareas l and r have same units.
- Area of a sector = ½ r²θ
- The sum of interior angles of a polygon of n sides = (n-2) × 180˚
  sin²x + cosec²x ≥ 2.
  cos²x + sec²x ≥ 2.
  tan²x + cot²x ≥ 2.
  Above are true for every real x.
- Maximum and minimum values of T-ratios:
i. If y = asinx + bcosx +c, then c – √ (a²+ b²) ≤ |y| ≤ c + √ (a²+ b²), Maximum value of y =c + √(a²+ b²)  Minimum value of y = c - √(a²+ b²)
ii. Greatest and least values of (asinx ± bcosx) are √ (a²+ b²) and √ (a² - b²) respectively.
iii. Maximum value of sinx and cosx = 1, minimum value = -1
iv. Max. value of (sinx.cosx) = 1/2 and min. value =-1/2.