Trigonometric Functions And Identities

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Discussion Overview

The discussion revolves around solving the equation tan²(x) = 5sec(x) - 2 within the interval 0 ≤ x ≤ 360. Participants explore different methods for solving the resulting quadratic equation derived from trigonometric identities.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant presents a problem involving the equation tan²(x) = 5sec(x) - 2 and attempts to solve it using the identity tan²(x) + 1 = sec²(x).
  • Another participant questions the necessity of transforming the equation into a quadratic form.
  • A participant explains their reasoning for using the identity to derive a quadratic equation, noting previous experiences with similar problems.
  • There is a suggestion that factoring may not be the only method for solving quadratic equations, prompting a discussion about alternative methods.
  • Completing the square is mentioned as a successful method for solving the quadratic equation by one participant.

Areas of Agreement / Disagreement

Participants express differing views on the approach to solving the equation, with some supporting the transformation into a quadratic and others questioning its necessity. The discussion remains unresolved regarding the best method to approach such problems.

Contextual Notes

Participants reference various methods for solving quadratic equations, including factoring, completing the square, and using the quadratic formula, but do not reach a consensus on the preferred approach for this specific problem.

wanchosen
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Hi there,

I am struggling to solve for x in the following problem:-

Find all values of x in the interval 0<= x <= 360 for which: tan^2(x) = 5sec(x) - 2

I have used the identity tan^2(x) + 1 = sec^2(x) to get:

sec^2(x) - 1 = 5sec(x) - 2

and rearranged to get

sec^2(x) - 5sec(x) + 1 = 0

but I can't factor this or solve for x. Can anyone help please?
 
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wanchosen said:
Can anyone help please?
Why did you spend all of that effort to turn it into a quadratic equation?
 
I assumed that I need to get all the trig functions of the same type before I can solve for x. Using the identity for sec^2(x) seemed the obvious choice...which lead me to a quadratic. I couldn't see how else to solve for x. I also tried converting everything in terms of cos (x) which was more messy. I may be mistaken, but from previous questions I have answered like this, tend to lead to a quadratic to factor for a solution/s for x.
 
wanchosen said:
tend to lead to a quadratic to factor for a solution/s for x.
Is factoring the only method you know for solving quadratic equations, or do you know other methods?
 
Completing the square, or using the quadratic formula are another two methods.
 
Thanks, tried completing the square to solve the quadratic and this worked.
 

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