Pre calc inverse help?

I'm confused about how to solve this problem. The answer says 1/2 but I don't know how they got that.

Sin[ sin^-1 1+ cos^-1 1/2)

6 Answers

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  • 1 month ago
    Favorite Answer

    sin⁻¹(1) → here, the goal is to find an angle where the sine is 1 → the angle is (π/2)

    cos⁻¹(1/2) → here, the goal is to find an angle where the cosine is 1/2 → the angle is (π/3)

    = sin[sin⁻¹(1) + cos⁻¹(1/2)]

    = sin[(π/2) + (π/3)]

    = sin[(3π/6) + (2π/6)]

    = sin(5π/6)

    = sin[(6π - π)/6]

    = sin[(6π/6) - (π/6)]

    = sin[π - (π/6)] → recall: sin(π - x) = sin(x)

    = sin(π/6)

    = 1/2

  • RockIt
    Lv 7
    1 month ago

    sin[ pi/2 + pi/3] = 1/2, by common triangle facts.

  • 1 month ago

    sin[sin^-1(1)+cos^-1(1/2)]

    =

    sin[sin^-1(1)cos[cos^-1(1/2)]+cos[sin^-1(1)]sin[cos^-1(1/2)]

    =

    1(1/2)+cos[pi/2]sin[pi/3]

    {for sin(pi/2)=1 & cos(pi/3)=1/2}

    =

    1/2+0*sin[pi/3]

    =

    1/2 (answer)

  • Vaman
    Lv 7
    1 month ago

    Sin[ sin^-1 1+ cos^-1 1/2)

    expand it sin (sin^-1 1) cos(cos^-1 1/2) + sin (cos^-1 1/2) cos (sin^-1 1)=1/2+ sin 60 cos 90

    =1/2,  cos 60 =1/2, 60=cos^-1 (1/2), sin 90=1

    90= sin^-1 1

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  • 1 month ago

    Let sin⁻¹(1) = θ

    sin(θ) = 1

    θ = 90°

    Let cos⁻¹(1/2) = ϕ

    cos(ϕ) = 1/2

    ϕ = 60° or ϕ = 300°

    Let y = sin[sin⁻¹(1) + cos⁻¹(1/2)]

    y = sin(θ + ϕ)

    y = sin(90° + 60°)   or  y = sin(90° + 300°)

    y = sin(150°)           or  y = sin(390°)

    y = sin(180° - 30°)  or  y = sin(360° - 30)

    y = sin(30°)

    y = 1/2

    Hence, sin[sin⁻¹(1) + cos⁻¹(1/2)] = 1/2

  • sin(a + b) = sin(a)cos(b) + sin(b)cos(a)

    sin(x)^2 + cos(x)^2 = 1

    sin(arcsin(t)) = t

    cos(arccos(t)) = t

    Using those identities, we can move forward

    sin(arcsin(1) + arccos(1/2)) =>

    sin(arcsin(1)) * cos(arccos(1/2)) + sin(arccos(1/2)) * cos(arcsin(1)) =>

    1 * (1/2) + sqrt(1 - cos(arccos(1/2))^2) * sqrt(1 - sin(arcsin(1))^2) =>

    (1/2) + sqrt(1 - (1/2)^2) * sqrt(1 - 1^2) =>

    (1/2) + sqrt(1 - 1/4) * sqrt(1 - 1) =>

    (1/2) + sqrt(3/4) * sqrt(0) =>

    (1/2) + (sqrt(3)/2) * 0 =>

    (1/2) + 0 =>

    (1/2)

    Or, we can find the angles for arcsin(1) and arccos(1/2)

    arcsin(1) = pi/2

    arccos(1/2) = pi/3

    pi/2 + pi/3 = 3pi/6 + 2pi/6 = 5pi/6

    sin(5pi/6) = 1/2

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