Problem (16) : Trigonometri


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  1. Jika x + y + z = 1800 dan tan x + tan y – 6 tan z = 1800. Buktikan bahwa tan x . tan y = 7

    PEMBAHASAN :

    tan x + tan y = 6 tan z

    tan x + tan y = 6 tan (1800 – (x + y))

    tan x + tan y = -6 tan (x + y)

    tan x + tan y = -6 \frac{tan \quad x+tan \quad y}{1-tan \quad x.tan \quad y}

    (tan x + tan y)(1 – tan x . tan y) = -6(tan x + tan y)

    1 – tan x . tan y = -6

    7 = tan x . tan y

  2. Buktikan \frac{sin(x-y)}{cos \quad x.cos \quad y} = tan x – tan y

    PEMBAHASAN :

    \frac{sin(x-y)}{cos(x).cos(y)} = \frac{sin(x).cos(y)-cos(x).sin(y)}{cos(x).cos(y)}

    = \frac{sin \quad x \quad cos \quad y}{cos \quad x \quad cos \quad y} \frac{cos \quad x \quad sin \quad y}{cos \quad x \quad cos \quad y}

    = \frac{sin \quad x}{cos \quad x} \frac{sin \quad y}{cos \quad y}

    = tan x – tan y

  3. Buktikan \frac{sin(x+y)+sin(x-y)}{cos(x+y)+cos(x-y)} = tan x

    PEMBAHASAN :

    \frac{sin(x+y)+sin(x-y)}{cos(x+y).cos(x-y)}

    = \frac{[sin(x).cos(y)+cos(x).sin(y)]+[sin(x).cos(y)-cos(x).sin(y)]}{[cos(x).cos(y)-sin(x).sin(y)]+[cos(x).cos(y)+sin(x).sin(y)]}

    = \frac{2.sin(x).cos(y)}{2.cos(x).cos(y)}

    = \frac{sin(x)}{cos(x)}

    = tan x

  4. Buktikan cos(x – y).cos (x + y) = cos2 x – sin2 y

    PEMBAHASAN :

    cos(x – y).cos (x + y) = [cos x cos y + sin x sin y].[cos x cos y – sin x sin y]

    = (cos x cos y)2 – (sin x sin y)2

    = cos2 x cos y2 – sin2 x sin y2

    NOTE : cos2 y + sin2 y = 1

    = cos2 x (1 – sin y2) – sin2 x (1 – cos y2)

    = cos2 x – cos2 y sin y2 – sin2 x + sin2 y cos y2

    = cos2 x – sin2 x

  5. Buktikan \frac{cos(x+y)}{sin(x-y)} = \frac{tan(x)+tan(y)}{1-tan(x).tan(y)}

    PEMBAHASAN :

    \frac{cos(x+y)}{sin(x-y)} = \frac{cos(x).cos(y)-sin(x).sin(y)}{sin(x).cos(y)-cos(x).sin(y)}

    = \frac{cos(x).cos(y)-sin(x).sin(y)}{sin(x).cos(y)-cos(x).sin(y)} x \frac{1/[cos(x).cos(y)]}{1/[cos(x).cos(y)]}

    = \frac{sin(x)/cos(y)+sin(x)/cos(y)}{1-sin(x).sin(y)/cos(x).cos(y)}

    = \frac{tan(x)+tan(y)}{1-tan(x).tan(y)}


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