Solve this trigonometric equation. 1 <msqrt> 2 </msqrt> </mfrac> ( sin

doturitip9 2022-07-02 Answered
Solve this trigonometric equation. 1 2 ( sin ( θ ) + cos ( θ ) ) = 1 2
Divide both sides by 1 2
sin ( θ ) + cos ( θ ) = 1
Divide both sides by cos ( θ )
tan(θ)+1=sec(θ)
square both sides:
tan ( θ ) + 1 = sec ( θ )
tan 2 ( θ ) + 2 tan ( θ ) + 1 = sec 2 ( θ )
Use the identity sec 2 ( θ ) = tan 2 ( θ ) + 1:
tan 2 ( θ ) + 2 tan ( θ ) + 1 = tan 2 ( θ ) + 1

2 tan ( θ ) = 0

tan ( θ ) = 0

θ = 0 , π , 2 π
I know that 0 and 2 π are correct but that π is wrong. I also know that the other correct answer is π 2 .
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Answers (1)

Freddy Doyle
Answered 2022-07-03 Author has 20 answers
The reason you got the extraneous solution θ = π is because you squared both sides of the equation tan θ + 1 = sec θ. You can check this by noting that tan π + 1 = 1 while sec π = 1, so θ = π is a solution to the squared equation but not the original. On the other hand, you missed out the solution θ = π / 2 because you divided by cos θ throughout, in which you've implicitly assumed cos θ 0 and hence θ π / 2
But these are rather easy to fix: check for extraneous solutions by substituting everything back into the original equation, and discuss the case cos θ = 0 (i.e. θ = π / 2) separately. Other than these two issues, your solution is perfect (and quite smart, actually).
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