Some Important Key Points

• The measure of an angle is the amount of rotation from the initial side of the terminal side.
• The sense of an angle is positive or negative according as the initial side rotates in anticlockwise or clockwise direction to get the terminal side.
• There are three systems of measuring angles:
          (a) Sexagesimal system
          (b) Centesimal system
          (c) Circular system
• In Sexagesimal system, we have

                  1 right angle = 90 degrees (90^∘)

                  1 degree        = 60 minute (60')

                  1 minute        = 60 seconds (60")

• In centesimal system, we have

                 1 right angle = 100 grades (100^g)

                 1 grade          = 100 minutes (100')

                 1 minute        = 100 seconds (100")

• In circular system, the unit of measurement is radian. One radian is the measure of an angle subtended at the centre of a circle by an arc of length equal to the radius of the circle

                      π radian = 180^∘

• The relation between three systems

                     D/90 degree = G/100 = 2R/π

• Six trigonometric ratios

                     sin θ, cos θ, cosec θ, sec θ, tan θ, cot θ

• 1 + tan² θ = sec² θ

       or  sec θ - tan θ = 1 / (sec θ + tan θ)

• 1 + cot² θ = cosec² θ

       or cosec θ - cot θ = 1 / (cosec θ + cot θ)

•                     sin (-θ) = - sin θ,
                     cos (-θ) = cos θ,
                    tan (- θ) = tan θ,
             sin (90^∘ - θ) = cos θ,
  and    cos (90^∘ - θ) = sin θ,
             sin (90^∘+ θ) = cos θ,
  and    cos (90^∘+ θ) = - sin θ,

• Sine and cosine functions and their reciprocals i.e., cosecant and secant functions are periodic functions with period 2π whereas tangent and cotangent functions are periodic with period π.
• Odd functions: Sine, tangent, cotangent, cosecant.
• Even functions: cosine, secant.
• The curve y = tan x is a symmetric in opposite quadrants and  -∞ < y < ∞  while the curve y = sec x is symmetric about y - axis and y ≥ 1 or y ≤ -1. the value of y do not exist for
                 x = (2x + 1)π/2
• For curve y = cosec x
  and           y = cot x, y do not exist for
                   x = nπ
• Formulae
              sin (A + B) = sin A cos B + cos A sin B
               sin (A - B) = sin A cos B - cos A sin B
             cos (A + B) = cos A cos B - sin A sin B
              cos (A - B) = cos A cos B + sin A sin B
             tan (A + B) = (tan A + tan B)/ (1 - tan A tan B)
              tan (A - B) = (tan A - tan B)/ (1 + tan A tan B)
  if                  A + B = π, then
                        sin A = sin B,
                        cos A = -cos B
  and               tan A = -tan B
  and if           A + B = 2π, then
                        sin A = -sin B,
                        cos A = cos B,
  and               tan A = -tan B
• The equation a cos θ + b sin θ = c is soluable for
          |c| ≤ √(a² + b²)
• In any ΔABC, we have
                a² = b² + c² - 2bc cos A
                b² = c² + a² - 2ca cos B
                c² = a² + b² - 2ab cos C
• In any ΔABC, we have
                a = b cos C + c cos B
                b = c cos A + a cos C
                c = a cos B + b cos A
• The area Δ of a ΔABC is given by
               Δ = (1/2) bc sin A
  or         Δ = (1/2) ca sin B
  or         Δ = (1/2) ab sin C
• Let P(n) be a statement involving the natural number n such that -
  (i) P(1) is true and (ii) P(m + 1) is true, whenever P(m) is true then, P(n) is true for all n ∈ N
  This is called first principle of mathematical inductions.
• Let P(n) be a statement involving the natural number n such that -
  (i) P(1) is true and (ii) P(m + 1) is true, whenever P(n) is true for all n ≤ m,  then, P(n) is true for all n ∈ N
  This is called second principle of mathematical inductions.
• Every polynomial equation f(x) = 0 of degree n has exactly n roots real or imaginary.
• If ax² + bx + c = 0, a ≠ 0 is a quadratic equation with real coefficients, then its roots α and β given by -
                         α = (-b + √(b² -4ac))/ (2a)
                         β = (-b - √(b² -4ac))/ (2a)
  or                   α = (-b + √D)/ (2a)
                         β = (-b - √D)/ (2a)
  where           D = b² -4ac
  (i) if              D = 0, then
                        α = β = -b/2a
  so, equation has real and equal roots.
  (ii) If  D = +ve and a perfect square, then roots are rational and unequal (for a, b, c ∈ ℚ)
  for a, b, c ∈ R, D = positive and perfect square, then roots are real and distinct.
  (iii) If D > 0, but not perfect square, then roots are irrational and unequal.
  (iv) If a = 1 and b, c ∈ I, then roots are rational numbers, and roots must be integer.
• n! = 1 × 2 × 3 × 4 . . . × (n - 1) × n
• (2n)!/(n)! = 1.3.5 . . .  (2n - 1)2n
• n! + 1 is not divisible by any natural number between 2 and n.