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.

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Properties of Triangle

1. Some results of conditional identity :
If A + B + C = π then
i. Sin2A + Sin2B + Sin2C = 4SinA.SinB.SinC
ii. SinA + SinB + SinC = 4CosA/2.CosB/2.CosC/2
iii. CosA + CosB + CosC = 1 + (4SinA/2.SinB/2.SinC/2)
iv. Sin²A + Sin²B + Sin²C = 2 + 2CosA.CosB.CosC
v. Cos2A/2 + Cos2B/2 + Cos2C/2 = 2 + 2SinA/2.SinB/2.SinC/2
vi. tanA + tanB + tanC = tanA.tanB.tanC
vii. tanA/2.tanB/2. + tanB/2.tanC/2 + tanC/2.tanA/2 = 1
viii. CotA/2.CotB/2 + CotB/2.CotC/2 + CotC/2.CotA/2 = 1

2.     Sine Law :

In any ΔABC
a/SinA = b/SinB = c/SinC = 2R (Radius of Circumference circle)
a = 2RSinA, SinA = a/2R
b = 2RSinB, SinB = b/2R
c = 2RSinC, SinC = c/2R

3.     Cosine Law :
In any Δ ABC,
CosA = [(b² + c² – a²)/2bc]
CosB = [(c² + a² – b²)/2ac]
CosC = [(a² + b² – c²)/2ab]

4. Projection law :
a = bcosA + cCosB
b = cCosA + aCosC
c = aCosA + bCosA

5. Tangent Law :
Tan[(B-C)/2] = [(b-c)/(b+c)]cot(A/2)]
Tan[(C-A)/2] = [(c-a)/(c+a)]cot(B/2)]
Tan[(A-B)/2] = [(a-b)/(a+b)]cot(C/2)]

6. Half angle formulae :
Here, S = (a + b + c)/2
SinA = √[{(s-b)(s-c)}/{bc}]
Sin(B/2) = √[{(s-a)(s-c)}/{ac}]
Sin(C/2) = √[{(s-a)(s-b)}/{ab}]
Cos(A/2) = √[{s(s-a)}/{bc}]
Cos(B/2) = √[{s(s-b)}/{ac}]
Cos(C/2) = √[{s(s-c)}/{ab}]
Tan(A/2) = √[{(s-b)(s-c)}/{s(s-a)}]
Tan(B/2) = √[{(s-a)(s-c)}/{s(s-b)}]
Tan(C/2) = √[{(s-a)(s-b)}/{s(s-c)}]
Cot(A/2) = √[{s(s-a)}/{(s-b)(s-c)}]
Cot(B/2) = √[{s(s-b)}/{(s-a)(s-c)}]
Cot(C/2) = √[{s(s-c)}/{(s-b)(s-a)}]

7. Area of triangle :
i. Δ =1/2.abSinC,
ii. Δ = √{s(s-a)(s-b)(s-c)}
iii. Δ = ¼√[2a²b² + 2b²c² + 2a²c² – a⁴ – b⁴ – c⁴]
iv. Δ = {abc}/{4R}
v. Δ = ½b.h
vi. Δ = √3/4.a2(for equilateral Δ)
Also,
tan(A/2)= [{(s-b)(s-c)}/Δ]= [Δ/{s(s-a)}]
Cot(A/2)= [{s(s-a)}/Δ]

8.     Formula for radii,

r →radius of in-circle.
i. r = Δ/s
ii. r = (s-a)tan(A/2)= (s-b)tan(B/2) = (s-c)tan(C/2)
iii. r = 4Rsin(A/2).sin(B/2).sin(C/2)

 

If r₁, r₂ & r₃ represents radius of ex-circles opposite to A, B, C then,
i.r₁ =∆/(s-a) r₂ = ∆/(s-b) r₃ = ∆/(s-c)
ii. r₁ = stan(A/2), r₂ = stan(B/2), r₃ = stan(C/2)
iii.r₁ = 4Rsin(A/2).cos(B/2).cos(C/2)
r₂ = 4Rcos(A/2).sin(B/2).cos(C/2)
r₃ = 4Rcos(A/2).cos(B/2).sin(C/2)

iv.r₁ = asec(A/2).cos(B/2).cos(C/2)
r₂ = acos(A/2).sec(B/2).cos(C/2)
r₃ = acos(A/2).cos(B/2).sec(C/2)

some important results are :
i. 4R = r₁+ r₂+ r₂- r
ii. 1/r₂ + 1/r₂ + 1/ r₃ = 1/r
iii. rr₁r₂r₃ = Δ²
iv. cosA + cosB + cosC = 1 + r/R
v. In an equilateral triangle R. r = 1/6.(side of the Δ²)
vi. If sinA, sinB, sinC are in A.P./G.P./H.P. then a, b, c are in A.P./G.P./H.P. respectively.
vii. If a, b, c are in H.P. then SinA, SinB, SinC are in A.P.
viii. If r₁, r₂, r₃ are in H.P. then sinA, sinB, sinC are in A.P.
ix. If cot(A/2), cot(B/2), cot(C/2)are in A.P. then a, b, c are in A.P.
x. If cotA, cotB, cotC are in A.P. then a², b², c² are in A.P.

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General Values and Trigonometric Equations

1. The trigonometric equation is an equation involving one or more trigonometric function of a variable. The equation may be true for one or more values, but not for every value of variable.
2. General Equation: The set of all possible solutions of trigonometric equation is called general solution of the equation.
3. Principal Value: The value of trigonometric function between 0 and 2π is known as Principal value.
4. The general solution of some trigonometric equation are–
a.
i. Sinθ = 0, θ = nπ
ii. Cosθ = 0, θ = (2n+1)π/2
iii. Tanθ = 0, θ = nπ
b.
i. Sinθ = Sinα, θ = nπ + (-1)ⁿα
ii. Cosecθ = Cosecα, θ = nπ + (-1)ⁿα
iii. Cosθ = Cosα, θ = 2nπ ± α
iv. Secθ = Secα , θ = 2nπ ± α
v. Tanθ = Tanα, θ = nπ + α
vi. Cotθ = Cotα , θ = nπ + α
c.
i. Sinθ = -Sinα, θ = nπ + (-1)ⁿ(-α)
ii. Cosθ = -Cosα, θ = 2nπ + (π-α)
iii. Tanθ = -Tanα, θ = nπ + (-α)
d.
i. Sin²θ = 1, θ = nπ + π/2
ii. Cos²θ = 1, θ = nπ
e.
i. Sin²θ = Sin²α, θ = nπ ± α
ii. Cos²θ = Cos²α, θ = nπ ± α
iii. Tan²θ = Tan²α , θ = nπ ± α
f.
i. Sinθ = 1, θ = (4n + 1)π/2
ii. Sinθ = -1, θ = (4n - 1)π/2
iii. Cosθ = 1, θ = 2nπ
iv. Cosθ = -1, θ = (2n+1)π
Note :
- Any value of x which makes both L.H.S. and R.H.S. equal will be a root but L.H.S. equal will be a root but the value of x for which ∞ = ∞ will not be solution as it is indeterminate form.
- When we square a given equation for finding solution, then, after finding roots, we have to check which root satisfies the origional equation.
- General values gives infinite solution.

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