Trigonometry


  1. If (sinα + cosecα)² + (cosα + secα)² = k + tan²α + cot²α, then the value of k is









  1. View Hint View Answer Discuss in Forum

    (sin α + cosec α)² + (cos α + sec α)² = k + tan²α + cot²α
    ⇒ sin²α + cosec²α + 2 sin α . cosec α + cos²α + sec²α + 2 cos α.sec α = k + tan²α + cot²α
    ⇒ sin²α + cos²α + 2 + cosec²α + sec²α + 2 = k + tan²α + cot²α
    ⇒ 5 + cosec²α + sec²α = k + tan²α + cot²α
    ⇒ 5 + 1 + cot²α + 1 + tan²α = k + tan²α + cot²α
    ⇒ 7 + cot²α + tan²α = k + tan²α + cot²α
    ⇒ k = 7

    Correct Option: B

    (sin α + cosec α)² + (cos α + sec α)² = k + tan²α + cot²α
    ⇒ sin²α + cosec²α + 2 sin α . cosec α + cos²α + sec²α + 2 cos α.sec α = k + tan²α + cot²α
    ⇒ sin²α + cos²α + 2 + cosec²α + sec²α + 2 = k + tan²α + cot²α
    ⇒ 5 + cosec²α + sec²α = k + tan²α + cot²α
    ⇒ 5 + 1 + cot²α + 1 + tan²α = k + tan²α + cot²α
    ⇒ 7 + cot²α + tan²α = k + tan²α + cot²α
    ⇒ k = 7


  1. If sin 21° =
    x
    , then sec 21° – sin 69° is equal to
    y









  1. View Hint View Answer Discuss in Forum

    sin 21° =
    x
    y

    cos 21° = √1 - sin² 21°

    ∴ sec 21° =
    y
    y² - x²

    ∴ sec 21° – sin 69°
    = sec 21° – sin (90° – 21°)
    = sec 21° – cos 21°
    =
    y
    -
    y² - x²
    y² - x²y

    =
    y² - (y² - x²)
    -
    y√y² - x²y√y² - x²

    Correct Option: A

    sin 21° =
    x
    y

    cos 21° = √1 - sin² 21°

    ∴ sec 21° =
    y
    y² - x²

    ∴ sec 21° – sin 69°
    = sec 21° – sin (90° – 21°)
    = sec 21° – cos 21°
    =
    y
    -
    y² - x²
    y² - x²y

    =
    y² - (y² - x²)
    -
    y√y² - x²y√y² - x²



  1. If sin
    πx
    = x² – 2x + 2, then the value of x is
    2









  1. View Hint View Answer Discuss in Forum

    sin
    πx
    = x² - 2x + 2
    2

    Putting x = 1
    sin
    π
    = 1 - 2 + 2 = 1
    2

    Correct Option: B

    sin
    πx
    = x² - 2x + 2
    2

    Putting x = 1
    sin
    π
    = 1 - 2 + 2 = 1
    2


  1. If tanθ – cotθ = 0, find the value of sinθ + cos θ.









  1. View Hint View Answer Discuss in Forum

    tanθ – cotθ = 0
    ⇒ tanθ = cotθ = tan (90° – q)
    ⇒ θ = 90° – θ ⇒ 2θ = 90°
    ⇒ θ = 45°
    ∴ sinθ + cosθ = sin 45° + cos 45°

    =
    1
    +
    1
    =
    2
    = √2
    222

    Correct Option: C

    tanθ – cotθ = 0
    ⇒ tanθ = cotθ = tan (90° – q)
    ⇒ θ = 90° – θ ⇒ 2θ = 90°
    ⇒ θ = 45°
    ∴ sinθ + cosθ = sin 45° + cos 45°

    =
    1
    +
    1
    =
    2
    = √2
    222



  1. The greatest value of sin4θ + cos4θ is









  1. View Hint View Answer Discuss in Forum

    cos4θ + sin4θ = (cos²θ + sin²θ)² – 2cos²θ sin²θ
    From maximum value, 2sin²θ.cos²θ = 0
    Hence, sin4θ + cos4θ = (1)² – 0 = 1

    Correct Option: D

    cos4θ + sin4θ = (cos²θ + sin²θ)² – 2cos²θ sin²θ
    From maximum value, 2sin²θ.cos²θ = 0
    Hence, sin4θ + cos4θ = (1)² – 0 = 1