Advertisements
Advertisements
Question
If `"sin"^-1 4/5 + "cos"^-1 12/13 = "sin"^-1 alpha`, then α = ______.
Options
`63/65`
`62/65`
`61/65`
`60/65`
Advertisements
Solution
If `"sin"^-1 4/5 + "cos"^-1 12/13 = "sin"^-1 alpha, "then" α = bbunderline(63/65)`.
Explanation:
Given: `"sin"^-1 4/5 + "cos"^-1 12/13 = "sin"^-1 alpha`
Step 1: Use identity for inverse cosine
`cos^-1 x = sin^-1 sqrt(1 - x^2)`
`cos^-1 (12/13) = sin^-1 (sqrt(1 - (12/13)^2)) = sin^-1 (5/13)`
Step 2: Now add the two inverse sines
`sin^-1 (4/5) + sin^-1 (5/13)`
`sin^-1 x + sin^-1 y = sin^-1 (xsqrt(1 - y^2) + ysqrt(1 - x^2)), "when" x^2 + y^2 ≤ 1`
Let x = `4/5, y = 5/13`
Now compute:
`sqrt(1 - y^2) = sqrt(1 - (5/13)^2) = sqrt(144/169) = 12/13`
`sqrt(1 - x^2) = sqrt(1 - (4/5)^2) = sqrt(9/25) = 3/5`
Now plug into the formula:
`alpha = 4/5 * 12/3 + 5/13 * 3/5`
`alpha = 48/65 + 15/65 = 63/65`
`alpha = 63/65`
APPEARS IN
RELATED QUESTIONS
Find the principal solution of the following equation:
Sec θ = `(2)/sqrt(3)`
Find the principal solution of the following equation:
`sqrt(3)` cosecθ + 2 = 0
Find the general solution of the following equation :
cosθ = `sqrt(3)/(2)`
Find the general solution of the following equation:
tan θ = - 1
Find the general solution of the following equation:
sin 2θ = `1/2`
Find the general solution of the following equation:
sin θ = tan θ
State whether the following equation has a solution or not?
2sinθ = 3
State whether the following equation have solution or not?
3 tanθ = 5
In Δ ABC, prove that a3 sin(B – C) + b3sin(C – A) + c3sin(A – B) = 0
Select the correct option from the given alternatives:
The principal solutions of equation sin θ = `- 1/2` are ______.
Select the correct option from the given alternatives:
If cos pθ = cos qθ, p ≠ q, then ______.
If `sqrt3 cos x − sin x = 1`, then general value of x is ______.
Select the correct option from the given alternatives:
In Δ ABC if ∠A = 45°, ∠B = 30°, then the ratio of its sides are
Select the correct option from the given alternatives:
In ΔABC if c2 + a2 – b2 = ac, then ∠B = ____.
If in a triangle, the angles are in A.P. and b: c = `sqrt3: sqrt2`, then A is equal to
Select the correct option from the given alternatives:
In any ΔABC, if acos B = bcos A, then the triangle is
Find the principal solutions of the following equation:
cot θ = 0
In ΔABC, prove that `("a - b")^2 cos^2 "C"/2 + ("a + b")^2 sin^2 "C"/2 = "c"^2`
If 2 tan-1(cos x) = tan-1(2 cosec x), then find the value of x.
Prove the following:
`cos^-1 "x" = tan^-1 (sqrt(1 - "x"^2)/"x")`, if x > 0
If | x | < 1, then prove that
`2 tan^-1 "x" = tan^-1 ("2x"/(1 - "x"^2)) = sin^-1 ("2x"/(1 + "x"^2)) = cos^-1 ((1 - "x"^2)/(1 + "x"^2))`
If `tan^-1 "x" + tan^-1 "y" + tan^-1 "z" = pi/2,` then show that xy + yz + zx = 1
Find the principal solutions of sin x = `-1/2`
If cos–1x + cos–1y – cos–1z = 0, then show that x2 + y2 + z2 – 2xyz = 1
If sin-1 x = `pi/10`, for some x ∈ [-1, 1], then the value of cos-1 x is _______.
`[sin (tan^-1 3/4)]^2 + [sin(tan^-1 4/3)]^2 = ?`
If x + y = `pi/2`, then the maximum value of sin x. sin y is.
The value of sin 18° is ______.
The value of θ in (π, 2π) satisfying the equation sin2θ - cos2θ = 1 is ______
The general solution of 4sin2 x = 1 is ______.
The general value of θ in the equation `2sqrt(3) cos theta = tan theta` is ______.
The equation 3sin2x + 10 cos x – 6 = 0 is satisfied, if ______.
Find the principal solutions of cot θ = 0
If 2 tan–1(cos x) = tan–1(2 cosec x). then find the value of x.
The general solution of the equation tan2 x = 1 is ______.
If a = sin θ + cos θ, b = sin3 θ + cos3 θ, then ______.
Find the general solution of sin θ + sin 3θ + sin 5θ = 0
Prove that the general solution of cos θ = cos α is θ = 2nπ ± α, n ∈ Z.
The general solution of cot 4x = –1 is ______.
If tan θ + sec θ = `sqrt(3)`, find the general value of θ.
If tan3θ = cotθ, then θ =
