Advertisements
Advertisements
प्रश्न
Prove the following:
`cos^-1(3/5) + cos^-1(4/5) = pi/(2)`
Advertisements
उत्तर
Let `cos^-1(3/5)` = x
∴ cos x = `(3)/(5), "where" 0 < x < pi/(2)`
∴ sin x > 0
Now,
sin x = `sqrt(1 - cos^2x)`
= `sqrt(1 - 9/25)`
= `sqrt(16/25)`
= `(4)/(5)`
∴ x = `sin^-1(4/5)`
∴ `cos^-1(3/5) = sin^-1(4/5)` ...(1)
L.H.S. = `cos^-1(3/5) + cos^-1(4/5)`
= `sin^-1(4/5) + cos^-1(4/5)` ...[By (1)]
= `pi/(2) ...[∵ sin^-1x + cos^-1x = pi/2]`
= R.H.S.
APPEARS IN
संबंधित प्रश्न
Find the principal value of the following:
`cos^(-1) (sqrt3/2)`
Find the principal value of the following:
`cos^(-1) (-1/sqrt2)`
If sin−1 x = y, then ______.
Find the domain of the following function:
`f(x)=sin^-1x^2`
Evaluate the following:
`tan^-1(tan (5pi)/6)+cos^-1{cos((13pi)/6)}`
Find the domain of `f(x)=cotx+cot^-1x`
Evaluate the following:
`cot^-1 1/sqrt3-\text(cosec)^-1(-2)+sec^-1(2/sqrt3)`
Evaluate: tan `[ 2 tan^-1 (1)/(2) – cot^-1 3]`
In ΔABC, if a = 18, b = 24, c = 30 then find the values of cos `A/2`
Find the principal value of the following: tan- 1( - √3)
Find the principal value of the following: sin-1 `(1/sqrt(2))`
Find the principal value of the following: cos- 1`(-1/2)`
Prove that sin `[tan^-1 ((1 - x^2)/(2x)) + cos^-1 ((1 - x^2)/(1 + x^2))]` = 1
Prove that cot−1(7) + 2 cot−1(3) = `pi/4`
Show that `tan^-1 (1/2) + tan^-1 (2/11) = tan^-1 (3/4)`
Solve: tan-1 (x + 1) + tan-1 (x – 1) = `tan^-1 (4/7)`
`sin^-1x + sin^-1 1/x + cos^-1x + cos^-1 1/x` = ______
The principle solutions of equation tan θ = -1 are ______
In Δ ABC, with the usual notations, if sin B sin C = `"bc"/"a"^2`, then the triangle is ______.
If `sin^-1x + cos^-1y = (3pi)/10,` then `cos^-1x + sin^-1y =` ______
The principal value of `tan^{-1(sqrt3)}` is ______
`sin^2(sin^-1 1/2) + tan^2 (sec^-1 2) + cot^2(cosec^-1 4)` = ______.
If sin `(sin^-1 1/3 + cos^-1 x) = 1`, then the value of x is ______.
If `3sin^-1((2x)/(1 + x^2)) - 4cos^-1((1 - x^2)/(1 + x^2)) + 2tan^-1((2x)/(1 - x^2)) = pi/3`, then x is equal to ______
`sin{tan^-1((1 - x^2)/(2x)) + cos^-1((1 - x^2)/(1 + x^2))}` is equal to ______
Show that `cos(2tan^-1 1/7) = sin(4tan^-1 1/3)`
All trigonometric functions have inverse over their respective domains.
`"cos" 2 theta` is not equal to ____________.
`"sin"^2 25° + "sin"^2 65°` is equal to ____________.
`("cos" 8° - "sin" 8°)/("cos" 8° + "sin" 8°)` is equal to ____________.
`"cos"^-1 1/2 + 2 "sin"^-1 1/2` is equal to ____________.
If tan-1 (x – 1) + tan-1 x + tan-1 (x + 1) = tan-1 3x, then the values of x are ____________.
If 6sin-1 (x2 – 6x + 8.5) = `pi`, then x is equal to ____________.
`"sin" ["cot"^-1 {"cos" ("tan"^-1 "x")}] =` ____________.
`"cos"^-1 ["cos" (2 "cot"^-1 (sqrt2 - 1))] =` ____________.
`"tan"^-1 sqrt3 - "sec"^-1 (-2)` is equal to ____________.
If a = `(2sin theta)/(1 + costheta + sintheta)`, then `(1 + sintheta - costheta)/(1 + sintheta)` is
what is the value of `cos^-1 (cos (13pi)/6)`
Let x = sin–1(sin8) + cos–1(cos11) + tan–1(tan7), and x = k(π – 2.4) for an integer k, then the value of k is ______.
Number of values of x which lie in [0, 2π] and satisfy the equation
`(cos x/4 - 2sinx) sinx + (1 + sin x/4 - 2cosx)cosx` = 0
If ax + b (sec (tan–1 x)) = c and ay + b (sec.(tan–1 y)) = c, then `(x + y)/(1 - xy)` = ______.
If y = `tan^-1 (sqrt(1 + x^2) - sqrt(1 - x^2))/(sqrt(1 + x^2) + sqrt(1 - x^2))`, then `dy/dx` is equal to ______.
If sin–1x – cos–1x = `π/6`, then x = ______.
Find the value of `tan^-1(x/y) + tan^-1((y - x)/(y + x))`
