📊 त्रिकोणमिति : महत्वपूर्ण सूत्र 📊
🍎 योग सूत्र
➭ Sin(A+B) = SinACosB+CosASinB
➭ Sin(A-B) = SinACosB-CosASinB
➭ Cos(A+B) = CosACosB-SinASinB
➭ Cos(A-B) = CosACosB+SinASinB
🍏 अन्तर सूत्र
➭ tan(A+B) = tanA+tanB/1-tanAtanB
➭ tan(A-B) = tanA-tanB/1+tanAtanB
🍎 C-D सूत्र
➭ SinC+SinD = 2Sin(C+D/2) Cos(C-D/2)
➭ SinC-SinD = 2Cos(C+D/2) Sin(C-D/2)
➭ CosC+CosD = 2Cos(C+D/2) Cos(C-D/2)
➭ CosC-CosD = 2Sin(C+D/2) Sin(D-C/2)
➭ CosC-CosD = -2Sin(C+D/2) Sin(C-D/2)
🍏 रूपांतरण सूत्र
➛ 2SinACosB = Sin(A+B)+Sin(A-B)
➛ 2CosASinB = Sin(A+B)-Sin(A-B)
➛ 2CosACosB = Cos(A+B)+Cos(A-B)
➛ 2SinASinB = Cos(A-B)-Cos(A+B)
🍎 द्विक कोण सूत्र
➛ Sin2A = 2SinACosA
➛ Cos2A = Cos²A-Sin²A = 2Cos²-1 = 1-2Sin²A
➛ tan2A = 2tanA/1-tan²A
➛ Sin2A = 2tanA/1+tan²A
➛ Cos2A = 1-tan²A/1+tan²A
🍏 विशिष्ट सूत्र
➛ Sin(A+B)Sin(A-B) = Sin²A-Sin²B
= Cos²B-Cos²A
➛ Cos(A+B)Cos(A-B) = Cos²A-Sin²B = Cos²B-Sin²A
🍎 त्रिक कोण सूत्र
➛ Sin3A = 3SinA-4Sin³A
➛ Cos3A = 4Cos³A-3CosA
➛ tan3A = 3tanA-tan³A/1-3tan²A
🍏 महत्वपूर्ण सर्वसमिकाएं
➛ Sin²θ+Cos²θ = 1
➭ Sin²θ = 1-Cos²θ
➭ Cos²θ = 1-Sin²θ
➛ 1+tan²θ = Sec²θ
➭ Sec²θ-tan²θ = 1
➭ tan²θ = Sec²θ-1
➛ 1+Cot²θ = Cosec²θ
➭ Cosec²θ-Cot²θ = 1
➭ Cot²θ = Cosec²θ-1
🍎 योग सूत्र
➭ Sin(A+B) = SinACosB+CosASinB
➭ Sin(A-B) = SinACosB-CosASinB
➭ Cos(A+B) = CosACosB-SinASinB
➭ Cos(A-B) = CosACosB+SinASinB
🍏 अन्तर सूत्र
➭ tan(A+B) = tanA+tanB/1-tanAtanB
➭ tan(A-B) = tanA-tanB/1+tanAtanB
🍎 C-D सूत्र
➭ SinC+SinD = 2Sin(C+D/2) Cos(C-D/2)
➭ SinC-SinD = 2Cos(C+D/2) Sin(C-D/2)
➭ CosC+CosD = 2Cos(C+D/2) Cos(C-D/2)
➭ CosC-CosD = 2Sin(C+D/2) Sin(D-C/2)
➭ CosC-CosD = -2Sin(C+D/2) Sin(C-D/2)
🍏 रूपांतरण सूत्र
➛ 2SinACosB = Sin(A+B)+Sin(A-B)
➛ 2CosASinB = Sin(A+B)-Sin(A-B)
➛ 2CosACosB = Cos(A+B)+Cos(A-B)
➛ 2SinASinB = Cos(A-B)-Cos(A+B)
🍎 द्विक कोण सूत्र
➛ Sin2A = 2SinACosA
➛ Cos2A = Cos²A-Sin²A = 2Cos²-1 = 1-2Sin²A
➛ tan2A = 2tanA/1-tan²A
➛ Sin2A = 2tanA/1+tan²A
➛ Cos2A = 1-tan²A/1+tan²A
🍏 विशिष्ट सूत्र
➛ Sin(A+B)Sin(A-B) = Sin²A-Sin²B
= Cos²B-Cos²A
➛ Cos(A+B)Cos(A-B) = Cos²A-Sin²B = Cos²B-Sin²A
🍎 त्रिक कोण सूत्र
➛ Sin3A = 3SinA-4Sin³A
➛ Cos3A = 4Cos³A-3CosA
➛ tan3A = 3tanA-tan³A/1-3tan²A
🍏 महत्वपूर्ण सर्वसमिकाएं
➛ Sin²θ+Cos²θ = 1
➭ Sin²θ = 1-Cos²θ
➭ Cos²θ = 1-Sin²θ
➛ 1+tan²θ = Sec²θ
➭ Sec²θ-tan²θ = 1
➭ tan²θ = Sec²θ-1
➛ 1+Cot²θ = Cosec²θ
➭ Cosec²θ-Cot²θ = 1
➭ Cot²θ = Cosec²θ-1
🔰Notes on Trigonometric Equations and Identities🔰
A function f(x) is said to be periodic if there exists some T > 0 such that f(x+T) = f(x) for all x in the domain of f(x).
In case, the T in the definition of period of f(x) is the smallest positive real number then this ‘T’ is called the period of f(x).
Periods of various trigonometric functions are listed below:
1) sin x has period 2π
2) cos x has period 2π
3) tan x has period π
4) sin(ax+b), cos (ax+b), sec(ax+b), cosec (ax+b) all are of period 2π/a
5) tan (ax+b) and cot (ax+b) have π/a as their period
6) |sin (ax+b)|, |cos (ax+b)|, |sec(ax+b)|, |cosec (ax+b)| all are of period π/a
7) |tan (ax+b)| and |cot (ax+b)| have π/2a as their period
Sum and Difference Formulae of Trigonometric Ratios
1) sin(a + ß) = sin(a)cos(ß) + cos(a)sin(ß)
2) sin(a – ß) = sin(a)cos(ß) – cos(a)sin(ß)
3) cos(a + ß) = cos(a)cos(ß) – sin(a)sin(ß)
4) cos(a – ß) = cos(a)cos(ß) + sin(a)sin(ß)
5) tan(a + ß) = [tan(a) + tan (ß)]/ [1 - tan(a)tan (ß)]
6)tan(a - ß) = [tan(a) - tan (ß)]/ [1 + tan (a) tan (ß)]
7) tan (π/4 + θ) = (1 + tan θ)/(1 - tan θ)
8) tan (π/4 - θ) = (1 - tan θ)/(1 + tan θ)
9) cot (a + ß) = [cot(a) . cot (ß) - 1]/ [cot (a) +cot (ß)]
10) cot (a - ß) = [cot(a) . cot (ß) + 1]/ [cot (ß) - cot (a)]
Double or Triple -Angle Identities
1) sin 2x = 2sin x cos x
2) cos2x = cos2x – sin2x = 1 – 2sin2x = 2cos2x – 1
3) tan 2x = 2 tan x / (1-tan 2x)
4) sin 3x = 3 sin x – 4 sin3x
5) cos3x = 4 cos3x – 3 cosx
6) tan 3x = (3 tan x - tan3x) / (1- 3tan 2x)
For angles A, B and C, we have
1) sin (A + B +C) = sinAcosBcosC + cosAsinBcosC + cosAcosBsinC - sinAsinBsinC
2) cos (A + B +C) = cosAcosBcosC- cosAsinBsinC - sinAcosBsinC - sinAsinBcosC
3) tan (A + B +C) = [tan A + tan B + tan C –tan A tan B tan C]/ [1- tan Atan B - tan B tan C –tan A tan C
4) cot (A + B +C) = [cot A cot B cot C – cotA - cot B - cot C]/ [cot A cot B + cot Bcot C + cot A cotC–1]
List of some other trigonometric formulas:
1) 2sinAcosB = sin(A + B) + sin (A - B)
2) 2cosAsinB = sin(A + B) - sin (A - B)
3) 2cosAcosB = cos(A + B) + cos(A - B)
4) 2sinAsinB = cos(A - B) - cos (A + B)
5) sin A + sin B = 2 sin [(A+B)/2] cos [(A-B)/2]
6) sin A - sin B = 2 sin [(A-B)/2] cos [(A+B)/2]
7) cosA + cos B = 2 cos [(A+B)/2] cos [(A-B)/2]
8) cosA - cos B = 2 sin [(A+B)/2] sin [(B-A)/2]
9) tanA ± tanB = sin (A ± B)/ cos A cos B
10)cot A ± cot B = sin (B ± A)/ sin A sin B
Method of solving a trigonometric equation:
1) If possible, reduce the equation in terms of any one variable, preferably x. Then solve the equation as you used to in case of a single variable.
2) Try to derive the linear/algebraic simultaneous equations from the given trigonometric equations and solve them as algebraic simultaneous equations.
3) At times, you might be required to make certain substitutions. It would be beneficial when the system has only two trigonometric functions.
Some results which are useful for solving trigonometric equations:
1) sin θ = sina and cosθ = cosa ⇒ θ = 2nπ + a
2) sin θ = 0 ⇒ θ = nπ
3) cosθ = 0 ⇒ θ = (2n + 1)π/2
4) tan θ = 0 ⇒ θ = nπ
5) sinθ = sina⇒ θ = nπ + (-1)na where a ∈ [–π/2, π/2]
6) cosθ= cos a ⇒ θ = 2nπ ± a, where a ∈[0,π]
7) tanθ = tana⇒ θ = nπ+ a, where a ∈[–π/2, π/2]
8) sinθ = 1 ⇒ θ= (4n + 1)π/2
9) sin θ = -1 ⇒ θ = (4n - 1) π /2
10) sin θ = -1 ⇒ θ = (2n +1) π /2
11) |sinθ| = 1⇒ θ =2nπ
12) cosθ = 1 ⇒ θ =(2n + 1)
13) |cosθ| = 1⇒ θ =nπ
A function f(x) is said to be periodic if there exists some T > 0 such that f(x+T) = f(x) for all x in the domain of f(x).
In case, the T in the definition of period of f(x) is the smallest positive real number then this ‘T’ is called the period of f(x).
Periods of various trigonometric functions are listed below:
1) sin x has period 2π
2) cos x has period 2π
3) tan x has period π
4) sin(ax+b), cos (ax+b), sec(ax+b), cosec (ax+b) all are of period 2π/a
5) tan (ax+b) and cot (ax+b) have π/a as their period
6) |sin (ax+b)|, |cos (ax+b)|, |sec(ax+b)|, |cosec (ax+b)| all are of period π/a
7) |tan (ax+b)| and |cot (ax+b)| have π/2a as their period
Sum and Difference Formulae of Trigonometric Ratios
1) sin(a + ß) = sin(a)cos(ß) + cos(a)sin(ß)
2) sin(a – ß) = sin(a)cos(ß) – cos(a)sin(ß)
3) cos(a + ß) = cos(a)cos(ß) – sin(a)sin(ß)
4) cos(a – ß) = cos(a)cos(ß) + sin(a)sin(ß)
5) tan(a + ß) = [tan(a) + tan (ß)]/ [1 - tan(a)tan (ß)]
6)tan(a - ß) = [tan(a) - tan (ß)]/ [1 + tan (a) tan (ß)]
7) tan (π/4 + θ) = (1 + tan θ)/(1 - tan θ)
8) tan (π/4 - θ) = (1 - tan θ)/(1 + tan θ)
9) cot (a + ß) = [cot(a) . cot (ß) - 1]/ [cot (a) +cot (ß)]
10) cot (a - ß) = [cot(a) . cot (ß) + 1]/ [cot (ß) - cot (a)]
Double or Triple -Angle Identities
1) sin 2x = 2sin x cos x
2) cos2x = cos2x – sin2x = 1 – 2sin2x = 2cos2x – 1
3) tan 2x = 2 tan x / (1-tan 2x)
4) sin 3x = 3 sin x – 4 sin3x
5) cos3x = 4 cos3x – 3 cosx
6) tan 3x = (3 tan x - tan3x) / (1- 3tan 2x)
For angles A, B and C, we have
1) sin (A + B +C) = sinAcosBcosC + cosAsinBcosC + cosAcosBsinC - sinAsinBsinC
2) cos (A + B +C) = cosAcosBcosC- cosAsinBsinC - sinAcosBsinC - sinAsinBcosC
3) tan (A + B +C) = [tan A + tan B + tan C –tan A tan B tan C]/ [1- tan Atan B - tan B tan C –tan A tan C
4) cot (A + B +C) = [cot A cot B cot C – cotA - cot B - cot C]/ [cot A cot B + cot Bcot C + cot A cotC–1]
List of some other trigonometric formulas:
1) 2sinAcosB = sin(A + B) + sin (A - B)
2) 2cosAsinB = sin(A + B) - sin (A - B)
3) 2cosAcosB = cos(A + B) + cos(A - B)
4) 2sinAsinB = cos(A - B) - cos (A + B)
5) sin A + sin B = 2 sin [(A+B)/2] cos [(A-B)/2]
6) sin A - sin B = 2 sin [(A-B)/2] cos [(A+B)/2]
7) cosA + cos B = 2 cos [(A+B)/2] cos [(A-B)/2]
8) cosA - cos B = 2 sin [(A+B)/2] sin [(B-A)/2]
9) tanA ± tanB = sin (A ± B)/ cos A cos B
10)cot A ± cot B = sin (B ± A)/ sin A sin B
Method of solving a trigonometric equation:
1) If possible, reduce the equation in terms of any one variable, preferably x. Then solve the equation as you used to in case of a single variable.
2) Try to derive the linear/algebraic simultaneous equations from the given trigonometric equations and solve them as algebraic simultaneous equations.
3) At times, you might be required to make certain substitutions. It would be beneficial when the system has only two trigonometric functions.
Some results which are useful for solving trigonometric equations:
1) sin θ = sina and cosθ = cosa ⇒ θ = 2nπ + a
2) sin θ = 0 ⇒ θ = nπ
3) cosθ = 0 ⇒ θ = (2n + 1)π/2
4) tan θ = 0 ⇒ θ = nπ
5) sinθ = sina⇒ θ = nπ + (-1)na where a ∈ [–π/2, π/2]
6) cosθ= cos a ⇒ θ = 2nπ ± a, where a ∈[0,π]
7) tanθ = tana⇒ θ = nπ+ a, where a ∈[–π/2, π/2]
8) sinθ = 1 ⇒ θ= (4n + 1)π/2
9) sin θ = -1 ⇒ θ = (4n - 1) π /2
10) sin θ = -1 ⇒ θ = (2n +1) π /2
11) |sinθ| = 1⇒ θ =2nπ
12) cosθ = 1 ⇒ θ =(2n + 1)
13) |cosθ| = 1⇒ θ =nπ
❄️Notes on Quadratic Equations⛄️
In order to solve a quadratic equation of the form ax2 + bx + c, we first need to calculate the discriminant with the help of the formula D = b2 – 4ac.
The solution of the quadratic equation ax2 + bx + c= 0 is given by x = [-b ± √ b2 – 4ac] / 2a
If α and β are the roots of the quadratic equation ax2 + bx + c = 0, then we have the following results for the sum and product of roots:
α + β = -b/a
α.β = c/a
α – β = √D/a
It is not possible for a quadratic equation to have three different roots and if in any case it happens, then the equation becomes an identity.
Nature of Roots:
Consider an equation ax2 + bx + c = 0, where a, b and c ∈ R and a ≠ 0, then we have the following cases:
D > 0 iff the roots are real and distinct i.e. the roots are unequal
D = 0 iff the roots are real and coincident i.e. equal
D < 0 iffthe roots are imaginary
The imaginary roots always occur in pairs i.e. if a+ib is one root of a quadratic equation, then the other root must be the conjugate i.e. a-ib, where a, b ∈ R and i = √-1.
Consider an equation ax2 + bx + c = 0, where a, b and c ∈Q and a ≠ 0, then
If D > 0 and is also a perfect square then the roots are rational and unequal.
If α = p + √q is a root of the equation, where ‘p’ is rational and √q is a surd, then the other root must be the conjugate of it i.e. β = p - √q and vice versa.
If the roots of the quadratic equation are known, then the quadratic equation may be constructed with the help of the formula
x2 – (Sum of roots)x + (Product of roots) = 0.
So if α and β are the roots of equation then the quadratic equation is
x2 – (α + β)x + α β = 0
For the quadratic expressiony = ax2 + bx + c, where a, b, c ∈ R and a ≠ 0, then the graph between x and y is always a parabola.
If a > 0, then the shape of the parabola is concave upwards
If a < 0, then the shape of the parabola is concave upwards
Inequalities of the form P(x)/ Q(x) > 0 can be easily solved by the method of intervals of number line rule.
The maximum and minimum values of the expression y = ax2 + bx + c occur at the point x = -b/2a depending on whether a > 0 or a< 0.
y ∈[(4ac-b2) / 4a, ∞] if a > 0
If a < 0, then y ∈ [-∞, (4ac-b2) / 4a]
The quadratic function of the form f(x, y) = ax2+by2 + 2hxy + 2gx + 2fy + c = 0 can be resolved into two linear factors provided it satisfies the following condition: abc + 2fgh –af2 – bg2 – ch2 = 0
In general, if α1,α2, α3, …… ,αn are the roots of the equation
f(x) = a0xn +a1xn-1 + a2xn-2 + ……. + an-1x + an, then
1.Σα1 = - a1/a0
2.Σ α1α2 = a2/a0
3.Σ α1α2α3 = - a3/a0
……… ……….
Σ α1α2α3 ……αn= (-1)n an/a0
Every equation of nth degree has exactly n roots (n ≥1) and if it has more than n roots then the equation becomes an identity.
If there are two real numbers ‘a’ and ‘b’ such that f(a) and f(b) are of opposite signs, then f(x) = 0 must have at least one real root between ‘a’ and ‘b’.
Every equation f(x) = 0 of odd degree has at least one real root of a sign opposite to that of its last term.
In order to solve a quadratic equation of the form ax2 + bx + c, we first need to calculate the discriminant with the help of the formula D = b2 – 4ac.
The solution of the quadratic equation ax2 + bx + c= 0 is given by x = [-b ± √ b2 – 4ac] / 2a
If α and β are the roots of the quadratic equation ax2 + bx + c = 0, then we have the following results for the sum and product of roots:
α + β = -b/a
α.β = c/a
α – β = √D/a
It is not possible for a quadratic equation to have three different roots and if in any case it happens, then the equation becomes an identity.
Nature of Roots:
Consider an equation ax2 + bx + c = 0, where a, b and c ∈ R and a ≠ 0, then we have the following cases:
D > 0 iff the roots are real and distinct i.e. the roots are unequal
D = 0 iff the roots are real and coincident i.e. equal
D < 0 iffthe roots are imaginary
The imaginary roots always occur in pairs i.e. if a+ib is one root of a quadratic equation, then the other root must be the conjugate i.e. a-ib, where a, b ∈ R and i = √-1.
Consider an equation ax2 + bx + c = 0, where a, b and c ∈Q and a ≠ 0, then
If D > 0 and is also a perfect square then the roots are rational and unequal.
If α = p + √q is a root of the equation, where ‘p’ is rational and √q is a surd, then the other root must be the conjugate of it i.e. β = p - √q and vice versa.
If the roots of the quadratic equation are known, then the quadratic equation may be constructed with the help of the formula
x2 – (Sum of roots)x + (Product of roots) = 0.
So if α and β are the roots of equation then the quadratic equation is
x2 – (α + β)x + α β = 0
For the quadratic expressiony = ax2 + bx + c, where a, b, c ∈ R and a ≠ 0, then the graph between x and y is always a parabola.
If a > 0, then the shape of the parabola is concave upwards
If a < 0, then the shape of the parabola is concave upwards
Inequalities of the form P(x)/ Q(x) > 0 can be easily solved by the method of intervals of number line rule.
The maximum and minimum values of the expression y = ax2 + bx + c occur at the point x = -b/2a depending on whether a > 0 or a< 0.
y ∈[(4ac-b2) / 4a, ∞] if a > 0
If a < 0, then y ∈ [-∞, (4ac-b2) / 4a]
The quadratic function of the form f(x, y) = ax2+by2 + 2hxy + 2gx + 2fy + c = 0 can be resolved into two linear factors provided it satisfies the following condition: abc + 2fgh –af2 – bg2 – ch2 = 0
In general, if α1,α2, α3, …… ,αn are the roots of the equation
f(x) = a0xn +a1xn-1 + a2xn-2 + ……. + an-1x + an, then
1.Σα1 = - a1/a0
2.Σ α1α2 = a2/a0
3.Σ α1α2α3 = - a3/a0
……… ……….
Σ α1α2α3 ……αn= (-1)n an/a0
Every equation of nth degree has exactly n roots (n ≥1) and if it has more than n roots then the equation becomes an identity.
If there are two real numbers ‘a’ and ‘b’ such that f(a) and f(b) are of opposite signs, then f(x) = 0 must have at least one real root between ‘a’ and ‘b’.
Every equation f(x) = 0 of odd degree has at least one real root of a sign opposite to that of its last term.
♦️Revision Notes on Arithmetic Progression♦️
If ‘a’ is the first term and ‘d’ is the common difference of the arithmetic progression, then its nth term is given by an = a+(n-1)d
The sum, Sn of the first ‘n’ terms of the A.P. is given by Sn = n/2 [2a + (n-1)d]
If Sn is the sum of n terms of an A.P. whose first term is ‘a’ and last term is ‘l’,Sn = (n/2)(a + l)
If common difference is d, number of terms n and the last term l, then Sn = (n/2)[2l-(n -1)d]
If a fixed number is added or subtracted from each term of an A.P., then the resulting sequence is also an A.P. and it has the same common difference as that of the original A.P.
If each term of A.P is multiplied by some constant or divided by a non-zero fixed constant, the resulting sequence is an A.P. again.
If a1, a2, a3, …, an andb1, b2, b3, …, bn, are in A.P. then a1+b1, a2+b2, a3+b3, ……, an+bn and a1–b1, a2–b2, a3–b3, ……, an–bn will also be in A.P.
Suppose a1, a2, a3, ……,an are in A.P. then an, an–1, ……, a3, a2, a1 will also be in A.P.
If nth term of a series is tn = An + B, then the series is in A.P.
If a1, a2, a3, ……, an are in A.P., then a1 + an = a2 + an–1 = a3 + an–2 = …… and so on.
In order to assume three terms in A.P. whose sum is given, they should be assumed as a-d, a, a+d.
Four terms of the A.P. whose sum is given should be assumed as a-3d, a-d, a+d, a+3d
Five convenient numbers in A.P. a–2b, a–b, a, a+b, a+2 b.
In general, we take a – rd, a – (r – 1)d, …., a – d, a, a + rd in case we have to take (2r + 1) terms in an A.P.
Likewise, any 2r terms of an A.P. should be assumed as: a – (2r-1)d, a – (2r – 3)d, …., a – d, a, a + d, ………….. , a+(2r-3)d, a + (2r-1)d.
The arithmetic mean of two numbers ‘a’ and ‘b’ is (a+b)/2.
The terms A1, A2, ….. , An are said to be arithmetic means between a and b if a, A1, A2, ….. , An, bis an A.P.
Clearly, ‘a’ is the first term, ‘b’ is the (n+2)th term and ‘d’ is the common difference. Then, we have b = a+(n+2-1)d = a+(n+1)d
Hence, this gives ‘d’ = (b-a)/(n+1)
If ‘a’ is the first term and ‘d’ is the common difference of the arithmetic progression, then its nth term is given by an = a+(n-1)d
The sum, Sn of the first ‘n’ terms of the A.P. is given by Sn = n/2 [2a + (n-1)d]
If Sn is the sum of n terms of an A.P. whose first term is ‘a’ and last term is ‘l’,Sn = (n/2)(a + l)
If common difference is d, number of terms n and the last term l, then Sn = (n/2)[2l-(n -1)d]
If a fixed number is added or subtracted from each term of an A.P., then the resulting sequence is also an A.P. and it has the same common difference as that of the original A.P.
If each term of A.P is multiplied by some constant or divided by a non-zero fixed constant, the resulting sequence is an A.P. again.
If a1, a2, a3, …, an andb1, b2, b3, …, bn, are in A.P. then a1+b1, a2+b2, a3+b3, ……, an+bn and a1–b1, a2–b2, a3–b3, ……, an–bn will also be in A.P.
Suppose a1, a2, a3, ……,an are in A.P. then an, an–1, ……, a3, a2, a1 will also be in A.P.
If nth term of a series is tn = An + B, then the series is in A.P.
If a1, a2, a3, ……, an are in A.P., then a1 + an = a2 + an–1 = a3 + an–2 = …… and so on.
In order to assume three terms in A.P. whose sum is given, they should be assumed as a-d, a, a+d.
Four terms of the A.P. whose sum is given should be assumed as a-3d, a-d, a+d, a+3d
Five convenient numbers in A.P. a–2b, a–b, a, a+b, a+2 b.
In general, we take a – rd, a – (r – 1)d, …., a – d, a, a + rd in case we have to take (2r + 1) terms in an A.P.
Likewise, any 2r terms of an A.P. should be assumed as: a – (2r-1)d, a – (2r – 3)d, …., a – d, a, a + d, ………….. , a+(2r-3)d, a + (2r-1)d.
The arithmetic mean of two numbers ‘a’ and ‘b’ is (a+b)/2.
The terms A1, A2, ….. , An are said to be arithmetic means between a and b if a, A1, A2, ….. , An, bis an A.P.
Clearly, ‘a’ is the first term, ‘b’ is the (n+2)th term and ‘d’ is the common difference. Then, we have b = a+(n+2-1)d = a+(n+1)d
Hence, this gives ‘d’ = (b-a)/(n+1)
🎓🎓🧩Mathematics formulas🧩🎯🎯
1. (α+в)²= α²+2αв+в²
2. (α+в)²= (α-в)²+4αв
3. (α-в)²= α²-2αв+в²
4. (α-в)²= (α+в)²-4αв
5. α² + в²= (α+в)² - 2αв.
6. α² + в²= (α-в)² + 2αв.
7. α²-в² =(α + в)(α - в)
8. 2(α² + в²) = (α+ в)² + (α - в)²
9. 4αв = (α + в)² -(α-в)²
10. αв =1. (α + в + ¢)² = α² + в² + ¢² + 2(αв + в¢ + ¢α)
12. (α + в)³ = α³ + 3α²в + 3αв² + в³
13. (α + в)³ = α³ + в³ + 3αв(α + в)
14. (α-в)³=α³-3α²в+3αв²-в³
15. α³ + в³ = (α + в) (α² -αв + в²)
16. α³ + в³ = (α+ в)³ -3αв(α+ в)
17. α³ -в³ = (α -в) (α² + αв + в²)
18. α³ -в³ = (α-в)³ + 3αв(α-в)
ѕιη0° =0
ѕιη30° = 1/2
ѕιη45° = 1/√2
ѕιη60° = √3/2
ѕιη90° = 1
¢σѕ ιѕ σρρσѕιтє σƒ ѕιη
тαη0° = 0
тαη30° = 1/√3
тαη45° = 1
тαη60° = √3
тαη90° = ∞
¢σт ιѕ σρρσѕιтє σƒ тαη
ѕє¢0° = 1
ѕє¢30° = 2/√3
ѕє¢45° = √2
ѕє¢60° = 2
ѕє¢90° = ∞
¢σѕє¢ ιѕ σρρσѕιтє σƒ ѕє¢
2ѕιηα¢σѕв=ѕιη(α+в)+ѕιη(α-в)
2¢σѕαѕιηв=ѕιη(α+в)-ѕιη(α-в)
2¢σѕα¢σѕв=¢σѕ(α+в)+¢σѕ(α-в)
2ѕιηαѕιηв=¢σѕ(α-в)-¢σѕ(α+в)
ѕιη(α+в)=ѕιηα ¢σѕв+ ¢σѕα ѕιηв.
» ¢σѕ(α+в)=¢σѕα ¢σѕв - ѕιηα ѕιηв.
» ѕιη(α-в)=ѕιηα¢σѕв-¢σѕαѕιηв.
» ¢σѕ(α-в)=¢σѕα¢σѕв+ѕιηαѕιηв.
» тαη(α+в)= (тαηα + тαηв)/ (1−тαηαтαηв)
» тαη(α−в)= (тαηα − тαηв) / (1+ тαηαтαηв)
» ¢σт(α+в)= (¢σтα¢σтв −1) / (¢σтα + ¢σтв)
» ¢σт(α−в)= (¢σтα¢σтв + 1) / (¢σтв− ¢σтα)
» ѕιη(α+в)=ѕιηα ¢σѕв+ ¢σѕα ѕιηв.
» ¢σѕ(α+в)=¢σѕα ¢σѕв +ѕιηα ѕιηв.
» ѕιη(α-в)=ѕιηα¢σѕв-¢σѕαѕιηв.
» ¢σѕ(α-в)=¢σѕα¢σѕв+ѕιηαѕιηв.
» тαη(α+в)= (тαηα + тαηв)/ (1−тαηαтαηв)
» тαη(α−в)= (тαηα − тαηв) / (1+ тαηαтαηв)
» ¢σт(α+в)= (¢σтα¢σтв −1) / (¢σтα + ¢σтв)
» ¢σт(α−в)= (¢σтα¢σтв + 1) / (¢σтв− ¢σтα)
α/ѕιηα = в/ѕιηв = ¢/ѕιη¢ = 2я
» α = в ¢σѕ¢ + ¢ ¢σѕв
» в = α ¢σѕ¢ + ¢ ¢σѕα
» ¢ = α ¢σѕв + в ¢σѕα
» ¢σѕα = (в² + ¢²− α²) / 2в¢
» ¢σѕв = (¢² + α²− в²) / 2¢α
» ¢σѕ¢ = (α² + в²− ¢²) / 2¢α
» Δ = αв¢/4я
» ѕιηΘ = 0 тнєη,Θ = ηΠ
» ѕιηΘ = 1 тнєη,Θ = (4η + 1)Π/2
» ѕιηΘ =−1 тнєη,Θ = (4η− 1)Π/2
» ѕιηΘ = ѕιηα тнєη,Θ = ηΠ (−1)^ηα
1. ѕιη2α = 2ѕιηα¢σѕα
2. ¢σѕ2α = ¢σѕ²α − ѕιη²α
3. ¢σѕ2α = 2¢σѕ²α − 1
4. ¢σѕ2α = 1 − ѕιη²α
5. 2ѕιη²α = 1 − ¢σѕ2α
6. 1 + ѕιη2α = (ѕιηα + ¢σѕα)²
7. 1 − ѕιη2α = (ѕιηα − ¢σѕα)²
8. тαη2α = 2тαηα / (1 − тαη²α)
9. ѕιη2α = 2тαηα / (1 + тαη²α)
10. ¢σѕ2α = (1 − тαη²α) / (1 + тαη²α)
11. 4ѕιη³α = 3ѕιηα − ѕιη3α
12. 4¢σѕ³α = 3¢σѕα + ¢σѕ3α
» ѕιη²Θ+¢σѕ²Θ=1
» ѕє¢²Θ-тαη²Θ=1
» ¢σѕє¢²Θ-¢σт²Θ=1
» ѕιηΘ=1/¢σѕє¢Θ
» ¢σѕє¢Θ=1/ѕιηΘ
» ¢σѕΘ=1/ѕє¢Θ
» ѕє¢Θ=1/¢σѕΘ
» тαηΘ=1/¢σтΘ
» ¢σтΘ=1/тαηΘ
» тαηΘ=ѕιηΘ/¢σѕΘ
1. (α+в)²= α²+2αв+в²
2. (α+в)²= (α-в)²+4αв
3. (α-в)²= α²-2αв+в²
4. (α-в)²= (α+в)²-4αв
5. α² + в²= (α+в)² - 2αв.
6. α² + в²= (α-в)² + 2αв.
7. α²-в² =(α + в)(α - в)
8. 2(α² + в²) = (α+ в)² + (α - в)²
9. 4αв = (α + в)² -(α-в)²
10. αв =1. (α + в + ¢)² = α² + в² + ¢² + 2(αв + в¢ + ¢α)
12. (α + в)³ = α³ + 3α²в + 3αв² + в³
13. (α + в)³ = α³ + в³ + 3αв(α + в)
14. (α-в)³=α³-3α²в+3αв²-в³
15. α³ + в³ = (α + в) (α² -αв + в²)
16. α³ + в³ = (α+ в)³ -3αв(α+ в)
17. α³ -в³ = (α -в) (α² + αв + в²)
18. α³ -в³ = (α-в)³ + 3αв(α-в)
ѕιη0° =0
ѕιη30° = 1/2
ѕιη45° = 1/√2
ѕιη60° = √3/2
ѕιη90° = 1
¢σѕ ιѕ σρρσѕιтє σƒ ѕιη
тαη0° = 0
тαη30° = 1/√3
тαη45° = 1
тαη60° = √3
тαη90° = ∞
¢σт ιѕ σρρσѕιтє σƒ тαη
ѕє¢0° = 1
ѕє¢30° = 2/√3
ѕє¢45° = √2
ѕє¢60° = 2
ѕє¢90° = ∞
¢σѕє¢ ιѕ σρρσѕιтє σƒ ѕє¢
2ѕιηα¢σѕв=ѕιη(α+в)+ѕιη(α-в)
2¢σѕαѕιηв=ѕιη(α+в)-ѕιη(α-в)
2¢σѕα¢σѕв=¢σѕ(α+в)+¢σѕ(α-в)
2ѕιηαѕιηв=¢σѕ(α-в)-¢σѕ(α+в)
ѕιη(α+в)=ѕιηα ¢σѕв+ ¢σѕα ѕιηв.
» ¢σѕ(α+в)=¢σѕα ¢σѕв - ѕιηα ѕιηв.
» ѕιη(α-в)=ѕιηα¢σѕв-¢σѕαѕιηв.
» ¢σѕ(α-в)=¢σѕα¢σѕв+ѕιηαѕιηв.
» тαη(α+в)= (тαηα + тαηв)/ (1−тαηαтαηв)
» тαη(α−в)= (тαηα − тαηв) / (1+ тαηαтαηв)
» ¢σт(α+в)= (¢σтα¢σтв −1) / (¢σтα + ¢σтв)
» ¢σт(α−в)= (¢σтα¢σтв + 1) / (¢σтв− ¢σтα)
» ѕιη(α+в)=ѕιηα ¢σѕв+ ¢σѕα ѕιηв.
» ¢σѕ(α+в)=¢σѕα ¢σѕв +ѕιηα ѕιηв.
» ѕιη(α-в)=ѕιηα¢σѕв-¢σѕαѕιηв.
» ¢σѕ(α-в)=¢σѕα¢σѕв+ѕιηαѕιηв.
» тαη(α+в)= (тαηα + тαηв)/ (1−тαηαтαηв)
» тαη(α−в)= (тαηα − тαηв) / (1+ тαηαтαηв)
» ¢σт(α+в)= (¢σтα¢σтв −1) / (¢σтα + ¢σтв)
» ¢σт(α−в)= (¢σтα¢σтв + 1) / (¢σтв− ¢σтα)
α/ѕιηα = в/ѕιηв = ¢/ѕιη¢ = 2я
» α = в ¢σѕ¢ + ¢ ¢σѕв
» в = α ¢σѕ¢ + ¢ ¢σѕα
» ¢ = α ¢σѕв + в ¢σѕα
» ¢σѕα = (в² + ¢²− α²) / 2в¢
» ¢σѕв = (¢² + α²− в²) / 2¢α
» ¢σѕ¢ = (α² + в²− ¢²) / 2¢α
» Δ = αв¢/4я
» ѕιηΘ = 0 тнєη,Θ = ηΠ
» ѕιηΘ = 1 тнєη,Θ = (4η + 1)Π/2
» ѕιηΘ =−1 тнєη,Θ = (4η− 1)Π/2
» ѕιηΘ = ѕιηα тнєη,Θ = ηΠ (−1)^ηα
1. ѕιη2α = 2ѕιηα¢σѕα
2. ¢σѕ2α = ¢σѕ²α − ѕιη²α
3. ¢σѕ2α = 2¢σѕ²α − 1
4. ¢σѕ2α = 1 − ѕιη²α
5. 2ѕιη²α = 1 − ¢σѕ2α
6. 1 + ѕιη2α = (ѕιηα + ¢σѕα)²
7. 1 − ѕιη2α = (ѕιηα − ¢σѕα)²
8. тαη2α = 2тαηα / (1 − тαη²α)
9. ѕιη2α = 2тαηα / (1 + тαη²α)
10. ¢σѕ2α = (1 − тαη²α) / (1 + тαη²α)
11. 4ѕιη³α = 3ѕιηα − ѕιη3α
12. 4¢σѕ³α = 3¢σѕα + ¢σѕ3α
» ѕιη²Θ+¢σѕ²Θ=1
» ѕє¢²Θ-тαη²Θ=1
» ¢σѕє¢²Θ-¢σт²Θ=1
» ѕιηΘ=1/¢σѕє¢Θ
» ¢σѕє¢Θ=1/ѕιηΘ
» ¢σѕΘ=1/ѕє¢Θ
» ѕє¢Θ=1/¢σѕΘ
» тαηΘ=1/¢σтΘ
» ¢σтΘ=1/тαηΘ
» тαηΘ=ѕιηΘ/¢σѕΘ
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