Solving an Equation in [-90,0]: Sec2A-3tanA-5=0

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To solve the equation sec²A - 3tanA - 5 = 0 for A in the interval [-90, 0], it is recommended to express sec²A in terms of tanA, using the identity sec²A = tan²A + 1. This substitution simplifies the equation into a quadratic form. The next step involves solving the resulting quadratic equation for tanA. Once tanA is determined, the corresponding angles A can be found within the specified interval. This method provides a clearer path to the solution.
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Homework Statement


Solve the Equation:

sec2A - 3tanA - 5=0, for A \in[-90,0]

Homework Equations





The Attempt at a Solution


\frac{1}{cos^{2}A} - \frac{3sinA}{cosA} -5 =0

\frac{1-3sinA.cosA}{cos^{2}A} -5 = 0

?
Did I go wrong? If not I'm stuck. Please help. Thank you.
 
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You didn't go wrong, but the form you wrote it in only makes it harder. I suggest you write the \sec^2 A term in terms of the tangent. Do you know how?
 
Last edited:
sec^2A=\frac{1}{cos^2A}=\frac{sin^2A+cos^2A}{cos^2A}=\frac{sin^2A}{cos^2A}+\frac{cos^2A}{cos^2A}=tan^2A+1

Just substitute and solve the quadratic equation.

Regards.
 
thanks
 

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