Ns Sidd Tutorial

HSC BOARD – IMPORTANT FORMULAS

MATHS IMPORTANT FORMULAS FOR ALL STUDENTS.

1) Here is a list of Algebraic formulas –

• a2 – b2 = (a – b)(a + b)
• (a+b)2 = a2 + 2ab + b2
• a2 + b2 = (a – b)2 + 2ab
• (a – b)2 = a2 – 2ab + b2
• (a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc
• (a – b – c)2 = a2 + b2 + c2 – 2ab – 2ac + 2bc
• (a + b)3 = a3 + 3a2b + 3ab2 + b3 ; (a + b)3 = a3 + b3 + 3ab(a + b)
• (a – b)3 = a3 – 3a2b + 3ab2 – b3
• a3 – b3 = (a – b)(a2 + ab + b2)
• a3 + b3 = (a + b)(a2 – ab + b2)
• (a + b)3 = a3 + 3a2b + 3ab2 + b3
• (a – b)3 = a3 – 3a2b + 3ab2 – b3
• (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4)
• (a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4)
• a4 – b4 = (a – b)(a + b)(a2 + b2)
• a5 – b5 = (a – b)(a4 + a3b + a2b2 + ab3 + b4)
• If n is a natural number an – bn = (a – b)(an-1 + an-2b+…+ bn-2a + bn-1)
• If n is even (n = 2k), an + bn = (a + b)(an-1 – an-2b +…+ bn-2a – bn-1)
• If n is odd (n = 2k + 1), an + bn = (a + b)(an-1 – an-2b +…- bn-2a + bn-1)
• (a + b + c + …)2 = a2 + b2 + c2 + … + 2(ab + ac + bc + ….)
• Laws of Exponents (am)(an) = am+n (ab)m = ambm (am)n = amn
• Fractional Exponents a0 = 1 aman=am−n am = 1a−m a−m = 1am

• Roots of Quadratic Equation

• • For a quadratic equation ax2 + bx + c where a ≠ 0, the roots will be given by the equation as −b±b2−4ac√2a
  • Δ = b2 − 4ac is called the discrimination
  • For real and distinct roots, Δ > 0
  • For real and coincident roots, Δ = 0
  • For non-real roots, Δ < 0
  • If α and β are the two roots of the equation ax2 + bx + c then, α + β = (-b / a) and α × β = (c / a).
  • If the roots of a quadratic equation are α and β, the equation will be (x − α)(x − β) = 0

• Factorials

• • n! = (1).(2).(3)…..(n − 1).n
  • n! = n(n − 1)! = n(n − 1)(n − 2)! = ….
  • 0! = 1
  • (a+b)n=an+nan−1b+n(n−1)2!an−2b2+n(n−1)(n−2)3!an−3b3+….+bn,where,n>1

Solved Examples

Question 1: Find out the value of 52 – 32Solution:
Using the formula a2 – b2 = (a – b)(a + b)
where a = 5 and b = 3
(a – b)(a + b)= (5 – 3)(5 + 3)= 2 × 8= 16
Question 2: 43 × 42 = ?
Solution:Using the exponential formula (am)(an) = am+nwhere a = 4
43 × 42= 43+2= 45= 1024

2) Here is a list of important Trigonometric formulas –

Quotient & Reciprocal Identities

tanθcscθ=sinθcosθcotθ=cosθsinθ=1sinθ secθ=1cosθ cotθ=1tanθ

Pythagorean Identities

sin2θ+cos2θ=11+cot2θ=csc2θtan2θ+1=sec2θ

Even & Odd Identities

sin(−x)csc(−x)=−sinx=−cscxcos(−x)=cosxsec(−x)=secxtan(−x)=−tanxcot(−x)=−cotx

Co-Function Identities

sin(π2−θ)=cosθcsc(π2−θ)=secθcos(π2−θ)=sinθsec(π2−θ)=cscθtan(π2−θ)=cotθcot(π2−θ)=tanθ

Sum and Difference Identities

cos(α+β)sin(α+β)tan(α+β)=cosαcosβ−sinαsinβ=sinαcosβ+cosαsinβ=tanα+tanβ1−tanαtanβcos(α−β)=cosαcosβ+sinαsinβsin(α−β)=sinαcosβ−cosαsinβtan(α−β)=tanα−tanβ1+tanαtanβ

Double Angle Identities

cos(2α)sin(2α)tan(2α)=cos2α−sin2α=2cos2α−1=1−2sin2α=2sinαcosβ=2tanα1−tan2α

Half Angle Identities

cosα2=±1+cosα2−−−−−−−−√sinα2=±1−cosα2−−−−−−−−√tanα2=1−cosαsinα=sinα1+cosα

Product to Sum & Sum to Product Identities

sina+sinbsina−sinbcosa+cosbcosa−cosb=2sina+b2cosa−b2=2sina−b2cosa+b2=2cosa+b2cosa−b2=2−2sina+b2sina−b2sinasinb=12[cos(a−b)−cos(a+b)]cosacosb=12[cos(a−b)+cos(a+b)]sinacosb=12[sin(a+b)+sin(a−b)]cosasinb=12[sin(a+b)−sin(a−b)]

Linear Combination Formula

Acosx+Bsinx=Ccos(x−D), where C=A2+B2−−−−−−−√,cosD=AC and sinD=BC

Review Questions

1. Find the sine, cosine, and tangent of an angle with terminal side on (−8,15).
2. If sina=5–√3 and tana<0, find seca.
3. Simplify: cos4x−sin4xcos2x−sin2x.
4. Verify the identity: 1+sinxcosxsinx=secx(cscx+1)

For problems 5-8, find all the solutions in the interval [0,2π).

5. sec(x+π2)+2=0
6. 8sin(x2)−8=0
7. 2sin2x+sin2x=0
8. 3tan22x=1
9. Solve the trigonometric equation 1−sinx=3–√sinx over the interval [0,π].
10. Solve the trigonometric equation 2cos3x−1=0 over the interval [0,2π].
11. Solve the trigonometric equation 2sec2x−tan4x=3 for all real values of x.

Find the exact value of:

12. cos157.5∘
13. sin13π12
14. Write as a product: 4(cos5x+cos9x)
15. Simplify: cos(x−y)cosy−sin(x−y)siny
16. Simplify: sin(4π3−x)+cos(x+5π6)
17. Derive a formula for sin6x.
18. If you solve cos2x=2cos2x−1 for cos2x, you would get cos2x=12(cos2x+1). This new formula is used to reduce powers of cosine by substituting in the right part of the equation for cos2x. Try writing cos4x in terms of the first power of cosine.
19. If you solve cos2x=1−2sin2x for sin2x, you would get sin2x=12(1−cos2x). Similar to the new formula above, this one is used to reduce powers of sine. Try writing sin4x in terms of the first power of cosine.
20. Rewrite in terms of the first power of cosine:
1. sin2xcos22x
2. tan42x

3) Here is a list of important Integration formulas –

The list of integral formulas are

• • ∫ 1 dx = x + C
  • ∫ a dx = ax+ C
  • ∫ x2 dx = ((xn+1)/(n+1))+C ; n≠1
  • ∫ sin x dx = – cos x + C
  • ∫ cos x dx = sin x + C
  • ∫ sec2 dx = tan x + C
  • ∫ csc2 dx = -cot x + C
  • ∫ sec x (tan x) dx = sec x + C
  • ∫ csc x ( cot x) dx = – csc x + C
  • ∫ (1/x) dx = ln |x| + C
  • ∫ ex dx = ex+ C
  • ∫ ax dx = (ax/ln a) + C ; a>0,  a≠1

These integral formulas are equally important as differentiation formulas. Some other important integration formulas are:

Classification of Integral Formulas

The above listed integral formulas are classified based on following functions,

• Rational functions
• Irrational functions
• Trigonometric functions
• Inverse trigonometric functions
• Hyperbolic functions
• Inverse hyperbolic functions
• Exponential functions
• Logarithmic functions
• Gaussian functions

Solve Using Integral Formulas

1. Calculate ∫ 5x4 dx

2. Find ∫x1+2x−−−−−√dx

3. Solve ∫1x2+6x+25dx