Trigonometry


  1. If sin α sec (30° + α) = 1 (0 < a < 60°), then the value of sin α + cos 2α is









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    sinα
    = 1
    cos(30° + α)

    sinα
    = 1
    sin(90° - 30 - α)

    sinα
    = 1
    sin(60° - α)

    ⇒ sin α = sin (60° – α)
    ⇒ 2α = 60° ⇒ α = 30°
    ∴ sinα + cos 2α
    = sin 30° + cos 60°
    =
    1
    +
    1
    = 1
    22

    Correct Option: A

    sinα
    = 1
    cos(30° + α)

    sinα
    = 1
    sin(90° - 30 - α)

    sinα
    = 1
    sin(60° - α)

    ⇒ sin α = sin (60° – α)
    ⇒ 2α = 60° ⇒ α = 30°
    ∴ sinα + cos 2α
    = sin 30° + cos 60°
    =
    1
    +
    1
    = 1
    22


  1. If cos4 θ – sin4 θ = (2 / 3) , then the value of 2 cos²θ– 1 is









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    cos4 θ - sin4 θ =
    2
    3

    (cos2 θ + sin2 θ) (cos2 θ – sin2 θ)
    2
    3

    ⇒ cos2 θ - sin2 θ
    2
    3

    ⇒ 2cos2 θ - 1
    2
    3

    Correct Option: C

    cos4 θ - sin4 θ =
    2
    3

    (cos2 θ + sin2 θ) (cos2 θ – sin2 θ)
    2
    3

    ⇒ cos2 θ - sin2 θ
    2
    3

    ⇒ 2cos2 θ - 1
    2
    3



  1. If sin α + cos β = 2 (0° ≤ β ≤ α ≤ 90°), then sin (2α + β / 3) =









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    sinα + cosβ = 2
    sinα ≤ 1 ; cosβ ≤ 1
    ⇒ α = 90° ; β = 0°

    = sin 60° =
    3
    2

    Also,
    cos
    α
    = cos 30° =
    3
    32

    Correct Option: B

    sinα + cosβ = 2
    sinα ≤ 1 ; cosβ ≤ 1
    ⇒ α = 90° ; β = 0°

    = sin 60° =
    3
    2

    Also,
    cos
    α
    = cos 30° =
    3
    32


  1. If A, B and C be the angles of a triangle, then put of the following, the incorrect relation is :









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    A + B + C = π

    A + B
    =
    π
    -
    C

    222

    ⇒ sin
    A + B
    2

    = sinπ-C
    22

    = cos
    C
    2

    Similarly,
    cosA + B = sinC
    22

    cotA + B = tanC
    22

    tanA + B = cotC
    22

    Correct Option: C

    A + B + C = π

    A + B
    =
    π
    -
    C

    222

    ⇒ sin
    A + B
    2

    = sinπ-C
    22

    = cos
    C
    2

    Similarly,
    cosA + B = sinC
    22

    cotA + B = tanC
    22

    tanA + B = cotC
    22



  1. If 0 < x < (π / 2) and secx = cosecy, then the value of sin (x + y) is :









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    secx = cosecy
    ⇒ cosx = siny

    ⇒ sin
    π
    - x = siny
    2

    ⇒ y =
    π
    - x
    2

    ⇒ x + y =
    π
    2

    ∴ sin (x + y) = sin
    π
    = 1
    2

    Correct Option: B

    secx = cosecy
    ⇒ cosx = siny

    ⇒ sin
    π
    - x = siny
    2

    ⇒ y =
    π
    - x
    2

    ⇒ x + y =
    π
    2

    ∴ sin (x + y) = sin
    π
    = 1
    2