if 3 sec²θ-5tan θ-4=0 find the general solution to this equation​

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if 3 sec²θ-5tan θ-4=0 find the general solution to this equation​

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MichaelMet 7 days 2021-07-16T07:20:09+00:00 1 Answers 7 views 0

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    2021-07-16T07:21:41+00:00

    3 sec²(θ) – 5 tan(θ) – 4 = 0

    Recall the Pythagorean identity,

    cos²(θ) + sin²(θ) = 1.

    Multiplying both sides by 1/cos²(θ) gives another form of the identity,

    1 + tan²(θ) = sec²(θ).

    Then the equation becomes quadratic in tan(θ):

    3 (1 + tan²(θ)) – 5 tan(θ) – 4 = 0

    3 tan²(θ) – 5 tan(θ) – 1 = 0

    I’ll solve by completing the square.

    tan²(θ) – 5/3 tan(θ)) – 1/3 = 0

    tan²(θ) – 5/3 tan(θ) = 1/3

    tan²(θ) – 5/3 tan(θ) + 25/36 = 1/3 + 25/36

    (tan(θ) – 5/6)² = 37/36

    tan(θ) – 5/6 = ±√37/6

    tan(θ) = (5 ± √37)/6

    Take the inverse tangent of both sides:

    θ = arctan((5 + √37)/6) +   or   θ = arctan((5 – √37)/6) +

    where n is any integer

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