Advertisements
Advertisements
Question
Prove that:
`cos^(-1) 12/13 + sin^(-1) 3/5 = sin^(-1) 56/65`
Advertisements
Solution
Let x = `cos^(-1) 12/13` and y = `sin^(-1) 3/5`
or cos x = `12/13` and sin y = `3/5`
sin x = `sqrt (1 - cos^2 x)` and cos y = `sqrt(1 - sin^2 y)`
Now, sin x = `sqrt(1 - 144/169)` and cos y = `sqrt( 1 - 9/25)`
⇒ sin x = `5/13` and cos y = `4/5`
We know that,
sin (x + y) = sin x cos y + cos x sin y
= `5/13 xx 4/5 + 12/13 xx 3/5 `
= `20/65 + 36/65 `
= `56/65`
⇒ x + y = `sin ^-1(56/65)`
or, `cos^-1(12/13) + sin^-1 (3/5)`
= `sin^-1(56/65)`
APPEARS IN
RELATED QUESTIONS
Prove that:
`tan^(-1)""1/5+tan^(-1)""1/7+tan^(-1)""1/3+tan^(-1)""1/8=pi/4`
Prove the following:
3cos−1x = cos−1(4x3 − 3x), `x ∈ [1/2, 1]`
Write the following function in the simplest form:
`tan^(-1) (sqrt((1-cos x)/(1 + cos x)))`, 0 < x < π
Write the following function in the simplest form:
`tan^(-1) ((3a^2 x - x^3)/(a^3 - 3ax^2)), a > 0; (-a)/sqrt3 < x < a/sqrt3`
Find the value of the following:
`tan 1/2 [sin^(-1) (2x)/(1+ x^2) + cos^(-1) (1-y^2)/(1+y^2)]`, |x| < 1, y > 0 and xy < 1
if `tan^(-1) (x-1)/(x - 2) + tan^(-1) (x + 1)/(x + 2) = pi/4` then find the value of x.
`sin[pi/3 - sin^(-1) (-1/2)]` is equal to ______.
Prove that:
`sin^(-1) 8/17 + sin^(-1) 3/5 = tan^(-1) 77/36`
Prove `(9pi)/8 - 9/4 sin^(-1) 1/3 = 9/4 sin^(-1) (2sqrt2)/3`
Solve `tan^(-1) - tan^(-1) (x - y)/(x+y)` is equal to
(A) `pi/2`
(B). `pi/3`
(C) `pi/4`
(D) `(-3pi)/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`
Find: ∫ sin x · log cos x dx
Find the value of the expression in terms of x, with the help of a reference triangle
sin (cos–1(1 – x))
Find the value of the expression in terms of x, with the help of a reference triangle
cos (tan–1 (3x – 1))
Find the value of `cot[sin^-1 3/5 + sin^-1 4/5]`
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)`
Simplify: `tan^-1 x/y - tan^-1 (x - y)/(x + y)`
Choose the correct alternative:
If |x| ≤ 1, then `2tan^-1x - sin^-1 (2x)/(1 + x^2)` is equal to
The value of cos215° - cos230° + cos245° - cos260° + cos275° is ______.
The value of `"tan"^ -1 (3/4) + "tan"^-1 (1/7)` is ____________.
`"cot" (pi/4 - 2 "cot"^-1 3) =` ____________.
If `"cot"^-1 (sqrt"cos" alpha) - "tan"^-1 (sqrt"cos" alpha) = "x",` the sinx is equal to ____________.
The value of cot `("cosec"^-1 5/3 + "tan"^-1 2/3)` is ____________.
The value of the expression tan `(1/2 "cos"^-1 2/sqrt3)`
If tan-1 2x + tan-1 3x = `pi/4,` then x is ____________.
If `"tan"^-1 (("x" - 1)/("x" + 2)) + "tan"^-1 (("x" + 1)/("x" + 2)) = pi/4,` then x is equal to ____________.
If x = a sec θ, y = b tan θ, then `("d"^2"y")/("dx"^2)` at θ = `π/6` is:
The value of `"tan"^-1 (1/2) + "tan"^-1(1/3) + "tan"^-1(7/8)` is ____________.
`"tan"^-1 1/3 + "tan"^-1 1/5 + "tan"^-1 1/7 + "tan"^-1 1/8 =` ____________.
If `"sin" {"sin"^-1 (1/2) + "cos"^-1 "x"} = 1`, then the value of x 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 value of `tan^-1 (x/y) - tan^-1 (x - y)/(x + y)` is equal to
`sin^-1(1 - x) - 2sin^-1 x = pi/2`, tan 'x' is equal to
`tan(2tan^-1 1/5 + sec^-1 sqrt(5)/2 + 2tan^-1 1/8)` is equal to ______.
Solve for x: `sin^-1(x/2) + cos^-1x = π/6`
If sin–1x + sin–1y + sin–1z = π, show that `x^2 - y^2 - z^2 + 2yzsqrt(1 - x^2) = 0`
