TRIGONOMETRIC FUNCTIONS

Size: px

Start display at page:

Download "TRIGONOMETRIC FUNCTIONS"

Nicholas Mason
6 years ago
Views:

Phone: -649 www.lirntlsses.in. Prove tht:. If sin θ = implies θ = nπ. If os θ = implies θ = (n+)π/. If tn θ = implies θ = nπ. Prove tht: MATHS I & II List of Theorems TRIGONOMETRIC FUNCTIONS.

1 Phone: Prove tht:. If sin θ = implies θ = nπ. If os θ = implies θ = (n+)π/. If tn θ = implies θ = nπ. Prove tht: MATHS I & II List of Theorems TRIGONOMETRIC FUNCTIONS. If sin θ = sin α implies θ = nπ + (-) n α. If os θ = os α implies θ = nπ±α. If θ nd α re not odd multiples of π/ then tn θ = tn α implies θ= nπ + α 3. Prove tht:. If sin θ=sin α implies θ = nπ±α. If os θ=os α implies θ = nπ±α. If tn θ=tn α implies θ = nπ±α 4. Prove tht os θ + sin θ = implies tht θ = nπ+α ±β where os α = +, sin α = + nd os β = + where,, ϵ R nd,, 5. Prove tht the sides of tringle re proportionl to the sines of the opposite ngles. (Sine rule) 6. In ny ΔABC prove tht (osine rule):. = + - os A. = + os B. = + os C 7. In ny ΔABC prove tht (projetion rule):. = os B + os C. = os C + os A. = os B + os A 8. In ny ΔABC prove tht, if + + = s then. sin A = (s )(s ). sin B = (s )(s ). sin C = (s )(s ) e. os B = s(s ) f. os C = s(s ) g. tn A = (s )(s ) s(s ) d. os A = s(s ) h. tn B = (s )(s ) s(s ) Corporte & Registered Offie: Unit No.3, st Floor, Alnkr CHS, Ndo Shopping Complex, Andheri (West), Mumi 458

2 Phone: i. tn C = (s )(s ) s(s ) 9. Prove tht re of ΔABC is given y A ABC = sin C = sin A = sin B. Prove Hero s Formul. PAIR OF LINES. Prove tht the joint eqution of pir of lines pssing through the origin is homogeneous eqution of degree two in x nd y.. Prove tht homogeneous eqution of degree two in x nd y represents pir of lines through the origin if h - 3. Prove tht the ute ngle etween the lines represented y x+hxy+y= is given y θ = tn h + VECTORS 4. Prove tht the two vetors nd re olliner if nd only if there exists slrs m nd n, t lest one of them is non-zero suh tht m + n = 5. Let nd e non olliner vetors. Prove tht vetor r is oplnr with nd if nd only if there exists unique slrs t, t suh tht r = t + t 6. Prove tht three vetors,, re oplnr if nd only if there exists non-zero liner omintion x +y +z = 7. If,, re three non oplnr vetors, then prove tht ny vetor r in the spe n e uniquely expressed s liner omintion of,, 8. Prove tht if A( ) nd B( )re ny two points in spe nd R(r ) e point on the line segment AB dividing it internlly in the rtio m:n then, r = m +n m+n 9. Prove tht if A( ) nd B( )re ny two points in spe nd R(r ) e point on the line segment AB dividing it externlly in the rtio m:n then, r = m n m n. Prove tht the volume of prllelopiped with o-terminus edges s,, is [ ]. Prove tht the volume of tetrhedron with o-terminus edges s,, is 6 [ ] 3D GEOMETRY. If l, m, n re diretion osines of line then prove tht l + m + n = 3. The ute ngle θ etween lines with diretion osines l,m,n nd l,m,n is given y os θ = l l +m m +n n Corporte & Registered Offie: Unit No.3, st Floor, Alnkr CHS, Ndo Shopping Complex, Andheri (West), Mumi 458

3 Phone: DERIVATIVES 4. If funtion f(x) is differentile t point, then it is lso ontinuous t tht point. 5. Chin Rule : If y = f(u) is differentile funtion of u nd u = g(x) is differentile funtion of x, then y = f(g(x)) is differentile funtion of x nd 6. Inverse Funtion : = du x du If y = f(x) is differentile funtion of x suh tht inverse funtion x = f (y) exists, then x is differentile funtion of y nd = ( ) 7. Derivtive of Inverse Funtion formule : where. If y = sin (x) nd x, π y π then = where x < x. If y = os (x) nd x, y π then = x where x <. If y = tn (x) nd x R, π y π then = + x d. If y = ot (x) nd x R, y π then = + x e. If y = se (x) nd x, y π nd y π then = ± x x f. If y = ose (x) nd x, π y π nd y then = x x 8. If x = f(t) nd y = g(t) re differentile funtions of prmeter t, then y is differentile funtion of x nd = dt where dt dt Corporte & Registered Offie: Unit No.3, st Floor, Alnkr CHS, Ndo Shopping Complex, Andheri (West), Mumi 458

4 Phone: INTEGRATION 9. If x = ϕ(t) is differentile funtion of t, then f(x) = f(ϕ(t)). ϕ (t) dt 3. Trigonometri Formule :. Prove tht tn x = log se x +. Prove tht : ot x = log sin x +. Prove tht : se x = log se x + tn x + d. Prove tht : ose x = log ose x ot x + 3. Polynomil Formule :. Prove tht : = x + tn ( x ) +. Prove tht : = x log +x x +. Prove tht : = log x x + x+ d. Prove tht : = x sin ( x ) + e. Prove tht : = log x + x + x + + f. Prove tht : = log x + x x + g. Prove tht : = x x se ( x ) + 3. If u nd v re two funtions of x then, 33. Prove tht : e x [f(x) + f (x)] = e x. f(x) Polynomil Formule in numertor : u. v = u v [ du v ]. x = x x + sin ( x ) +. x + = x x + + log x + x + +. x = x x log x + x + Corporte & Registered Offie: Unit No.3, st Floor, Alnkr CHS, Ndo Shopping Complex, Andheri (West), Mumi 458

5 Phone: Properties :. f(x) = f(x). f(x) = f(t) dt DEFINITE INTEGRAL. f(x) = f(x) + f(x) where < < d. f(x) = f( x) e. f(x) = f( + x) f. f(x) = f(x) + f( x) g. f(x) = f(x) if f(x) is n even funtion = if f(x) is n odd funtion ******* Corporte & Registered Offie: Unit No.3, st Floor, Alnkr CHS, Ndo Shopping Complex, Andheri (West), Mumi 458

Topics Covered: Pythagoras Theorem Definition of sin, cos and tan Solving right-angle triangles Sine and cosine rule

Topics Covered: Pythagoras Theorem Definition of sin, cos and tan Solving right-angle triangles Sine and cosine rule Trigonometry Topis overed: Pythgors Theorem Definition of sin, os nd tn Solving right-ngle tringles Sine nd osine rule Lelling right-ngle tringle Opposite (Side opposite the ngle θ) Hypotenuse (Side opposite