Advertisements
Advertisements
प्रश्न
Prove that `cos^(-1) 12/13 + sin^(-1) 3/5 = sin^(-1) 56/65`.
Advertisements
उत्तर
Let `sin^-1 3/5 = x`.
Then, `sin x = 3/5`
⇒ `cos x = sqrt(1 - (3/5)^2`
= `sqrt(16/25)`
= `4/5`
∴ `tan x = 3/4` ⇒ `x = tan^-1 3/4`
∴ `sin^-1 3/5 = tan^-1 3/4` ...(1)
Now, let `cos^-1 12/13 = y`.
Then, `cos y = 12/13` ⇒ `sin y = 5/13`.
∴ `tan y = 5/12` ⇒ `y = tan^-1 5/12`
∴ `cos^-1 12/13 = tan^-1 5/12` ...(2)
Let `sin^-1 56/65 = z`.
Then, `sin z = 56/65` ⇒ `cos z = 33/65`.
∴ `tan z = 56/33` ⇒ `z = tan^-1 56/33`
∴ `sin^-1 56/65 = tan^-1 56/33` ...(3)
Now, we have:
L.H.S. = `cos^-1 12/13 + sin^-1 3/5`
= `tan^-1 5/12 + tan^-1 3/4` ...[Using (1) and (2)]
= `tan^-1 (5/12 + 3/4)/(1 - 5/12 * 3/4)` ...`[tan^-1x + tan^-1y = tan^-1 (x + y)/(1 - xy)]`
= `tan^-1 (20 + 36)/(48 - 15)`
= `tan^-1 56/33`
= `sin^-1 56/65` = R.H.S. ...[Using (3)]
APPEARS IN
संबंधित प्रश्न
Prove that `tan^(-1)((6x-8x^3)/(1-12x^2))-tan^(-1)((4x)/(1-4x^2))=tan^(-1)2x;|2x|<1/sqrt3`
Write the following function in the simplest form:
`tan^(-1) (sqrt(1 + x^2) - 1)/x, x ≠ 0`
Write the function in the simplest form: `tan^(-1) 1/(sqrt(x^2 - 1)), |x| > 1`
Prove that `sin^(-1) 8/17 + sin^(-1) 3/5 = tan^(-1) 77/36`.
Prove that `tan^(-1) sqrt(x) = 1/2 cos^(-1) (1 - x)/(1 + x), x ∈ [0, 1]`.
Prove that `cot^(-1) ((sqrt(1 + sin x) + sqrt(1 - sinx))/(sqrt(1 + sin x) - sqrt(1 - sinx))) = x/2, x ∈ (0, pi/4)`.
sin–1 (1 – x) – 2 sin–1 x = `pi/2`, then x is equal to ______.
Solve `tan^(-1) - tan^(-1) (x - y)/(x+y)` is equal to
(A) `pi/2`
(B). `pi/3`
(C) `pi/4`
(D) `(-3pi)/4`
Solve the following equation for x: `cos (tan^(-1) x) = sin (cot^(-1) 3/4)`
If cos-1 x + cos -1 y + cos -1 z = π , prove that x2 + y2 + z2 + 2xyz = 1.
If y = `(x sin^-1 x)/sqrt(1 -x^2)`, prove that: `(1 - x^2)dy/dx = x + y/x`
Solve for x : `tan^-1 ((2-"x")/(2+"x")) = (1)/(2)tan^-1 ("x")/(2), "x">0.`
Find the value of `sin^-1[cos(sin^-1 (sqrt(3)/2))]`
Prove that `tan^-1 2/11 + tan^-1 7/24 = tan^-1 1/2`
Prove that `tan^-1x + tan^-1y + tan^-1z = tan^-1[(x + y + z - xyz)/(1 - xy - yz - zx)]`
Prove that `tan^-1x + tan^-1 (2x)/(1 - x^2) = tan^-1 (3x - x^3)/(1 - 3x^2), |x| < 1/sqrt(3)`
Solve: `2tan^-1 (cos x) = tan^-1 (2"cosec" x)`
Choose the correct alternative:
`sin^-1 (tan pi/4) - sin^-1 (sqrt(3/x)) = pi/6`. Then x is a root of the equation
Choose the correct alternative:
The equation tan–1x – cot–1x = `tan^-1 (1/sqrt(3))` has
Solve the equation `sin^-1 6x + sin^-1 6sqrt(3)x = - pi/2`
If 3 tan–1x + cot–1x = π, then x equals ______.
If |x| ≤ 1, then `2 tan^-1x + sin^-1 ((2x)/(1 + x^2))` is equal to ______.
The number of real solutions of the equation `sqrt(1 + cos 2x) = sqrt(2) cos^-1 (cos x)` in `[pi/2, pi]` is ______.
If y = `2 tan^-1x + sin^-1 ((2x)/(1 + x^2))` for all x, then ______ < y < ______.
The maximum value of sinx + cosx is ____________.
The value of the expression tan `(1/2 "cos"^-1 2/sqrt3)`
`"cot" ("cosec"^-1 5/3 + "tan"^-1 2/3) =` ____________.
The value of cot-1 9 + cosec-1 `(sqrt41/4)` is given by ____________.
The value of `"tan"^-1 (1/2) + "tan"^-1(1/3) + "tan"^-1(7/8)` is ____________.
`"sin"^-1 ((-1)/2)`
The Government of India is planning to fix a hoarding board at the face of a building on the road of a busy market for awareness on COVID-19 protocol. Ram, Robert and Rahim are the three engineers who are working on this project. “A” is considered to be a person viewing the hoarding board 20 metres away from the building, standing at the edge of a pathway nearby. Ram, Robert and Rahim suggested to the firm to place the hoarding board at three different locations namely C, D and E. “C” is at the height of 10 metres from the ground level. For viewer A, the angle of elevation of “D” is double the angle of elevation of “C” The angle of elevation of “E” is triple the angle of elevation of “C” for the same viewer. Look at the figure given and based on the above information answer the following:
Measure of ∠DAB = ________.
The Government of India is planning to fix a hoarding board at the face of a building on the road of a busy market for awareness on COVID-19 protocol. Ram, Robert and Rahim are the three engineers who are working on this project. “A” is considered to be a person viewing the hoarding board 20 metres away from the building, standing at the edge of a pathway nearby. Ram, Robert and Rahim suggested to the firm to place the hoarding board at three different locations namely C, D and E. “C” is at the height of 10 metres from the ground level. For viewer A, the angle of elevation of “D” is double the angle of elevation of “C” The angle of elevation of “E” is triple the angle of elevation of “C” for the same viewer. Look at the figure given and based on the above information answer the following:
Measure of ∠EAB = ________.
The Government of India is planning to fix a hoarding board at the face of a building on the road of a busy market for awareness on COVID-19 protocol. Ram, Robert and Rahim are the three engineers who are working on this project. “A” is considered to be a person viewing the hoarding board 20 metres away from the building, standing at the edge of a pathway nearby. Ram, Robert and Rahim suggested to the firm to place the hoarding board at three different locations namely C, D and E. “C” is at the height of 10 metres from the ground level. For viewer A, the angle of elevation of “D” is double the angle of elevation of “C” The angle of elevation of “E” is triple the angle of elevation of “C” for the same viewer. Look at the figure given and based on the above information answer the following:
𝐴' Is another viewer standing on the same line of observation across the road. If the width of the road is 5 meters, then the difference between ∠CAB and ∠CA'B is ______.
The Government of India is planning to fix a hoarding board at the face of a building on the road of a busy market for awareness on COVID-19 protocol. Ram, Robert and Rahim are the three engineers who are working on this project. “A” is considered to be a person viewing the hoarding board 20 metres away from the building, standing at the edge of a pathway nearby. Ram, Robert and Rahim suggested to the firm to place the hoarding board at three different locations namely C, D and E. “C” is at the height of 10 metres from the ground level. For viewer A, the angle of elevation of “D” is double the angle of elevation of “C” The angle of elevation of “E” is triple the angle of elevation of “C” for the same viewer. Look at the figure given and based on the above information answer the following:
Domain and Range of tan-1 x = ________.
The Simplest form of `cot^-1 (1/sqrt(x^2 - 1))`, |x| > 1 is
Find the value of `sin^-1 [sin((13π)/7)]`
The set of all values of k for which (tan–1 x)3 + (cot–1 x)3 = kπ3, x ∈ R, is the internal ______.
