Measurement of angel’s : There are two common systems of measuring angel’s
1.sexagesimal measure : According to this system, divide a right angle 90 into equal parts called degrees. Each degree is divided into sixty equal parts called minutes and each minute is divided into sixty equal parts called seconds. Denoting one degree, one minute and one second by the symbols 1^{0 }1’1”.
2. Circular measure : The unit of measurement is called radian . ∏ is an irrational number. Its approximate value is 22/7. Its value correct to 5 decimal places is 3.14159. Sometimes1/∏ value also required & is given by 0.31831.
Relation between degree & radian {1 radian = 180^{0} x 1/∏ = 57^{0}17’44.8”}; 1^{0} = ∏ /180 radian
Formulas from Trigonometry:
1. sin^{2}A + cos^{2}A = 1 ; sin^{2}A = 1- cos^{2}A ; cos^{2}A = 1- sin^{2}A
2. 1+tan^{2}A = sec^{2}A ; tan^{2}A = sec^{2}A -1 ; tan^{2}A - sec^{2}A = 1
3. sin(A+B) = sin A . cos B + cos A . sin B
4. sin(A - B) = sin A . cos B - cos A . sin B
5. cos(A+B) = cos A . cos B – sin A . sin B
6. cos(A - B) = cos A . cos B + sin A . sin B
7. sin(A+B) . sin(A - B) = sin^{2}A-sin^{2}B = cos^{2}B- cos^{2}A
8. cos(A+B) . cos(A - B) = cos^{2}A – sin^{2}B = cos^{2}B-sin^{2}A
9. sin2θ = 2 sinθ cosθ = 2tan θ / 1+tan^{2}θ
10. cos2θ = cos^{2}θ – sin^{2}θ =1- tan^{2}θ / 1+tan^{2}θ = 1-2 Sin^{2}θ =2cos^{2}θ -1
Formulas from Differentiation:
1. y = x^{n}, dy/dx = nx^{n-1}
2. y = e^{x}, dy/dx = e^{x}
3. y = log x, dy/dx = 1/x
4. y = a^{x}, dy/dx = a^{x} log a
5. y = sinx , dy/dx = cos x
6. y = cosx, dy/dx = - sinx
Scalars & Vectors
Those quantities which have only magnitude and are not related to any direction in space are called scalars. Whereas those which have both magnitude and direction are called vector quantities.
1. Vectors are said to be like vectors when they have the same sense of direction, unlike vectors when they are in opposite direction.
2. When two or more vectors are said to be collinear vectors when they act along the same line of action & parallel vectors when they are in parallel lines
3. Three or more vectors are said to be coplanar when they are parallel to the same plane or lie in the same plane whatever their magnitudes be (**two vectors are always coplanar)
4. Two vectors are said to be equal when they have the same length(magnitude &are parallel having the same sense of direction)are called as equal vectors.
5. If the origin and terminal points of a vector coincide, then it is said to be a zero vector
6. { Unit vector = Force vector / Magnitude of vector}
7. The vectors having the same initial point are call co –initial vectors.
8. A vector drawn parallel to a given vector through a specified point in space is called a localized vector. But if the origin of vectors is not specified the vectors are called free vectors.
Formulas from Vectors
1. a.b = b.a = ab cosθ
if the scalar product is of two vectors is zero, then at least one of the vectors is a zero vector or they are perpendicular.
2. a.(b+c) = a.b + a.c
3. i^{2}=j^{2}=k^{2}=1
4. i.j=j.k=k.i=0
5. a x b ≠ b x a but a x b = - b x a
6. i x i = j x j = k x k =0
7. i x j = k = - j x i ; j x k = i = - k x j; k x i = - i x k
8. if a = a_{1}i+a_{2}j+a_{3}j ; b = b_{1}i+b_{2}j+b_{3}k then a x b is given by
a x b = (a_{2}b_{3} – a_{3}b_{2}) i +(a_{3}b_{1}-a_{1}b_{3}) j + (a_{1}b_{2} – a_{2}b_{1}) k
Conversion Equivalents:
Length 1in = 2.540cm ; 1ft =12in =30.48cm
1mile=5280ft=1.609km
Force 1lb= 0.4536kg = 4.448N
Velocity 88fps=60mph=96.54km/hr
Acceleration g = 32.2fps^{2} = 9.81m/s^{2}
Pressure 1atm = 14.7psi=760mmof Hg= 1.013 x 10^{5} N/m2
Volume 1cuft=7.481gallons=28.32litres
Sign Conventions:
| I Quadrant | II Quadrant | III Quadrant | IV Quadrant |
X -Axis | + | - | - | + |
y-Axis | + | + | - | - |
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