Trigonometric equations that involve new ranges

[Vikash]

My question is solve tan(45-θ) = -1 in the range 0≤θ≤360 degrees but when I solve it normally and find the value of tan^-1(-1) I get -45^o. And no problem until that point. But the new range I get is 45^o≤45-θ,≤-315^o which cannot be solved therefore my dought is after this step WHY do you switch the sign of the range and have the new range as 45≥45-θ≥ -315 which thereafter iam okay....why do you change the inequality sign...is it because theta is being multiplied by a negative number (-1) so we have to switch the sign or is it some other explanation ?? HELP IS REALLY APPRECIATED ...THANKS.

 

[Cubist]

You start asking about tan(45-θ) = -1, but I think your question is really just about simplifying/ manipulating range definitions. Therefore I'll try to help with the latter, and if you need further help with "the bigger, original, question" then please post back.

0 ≤ θ ≤ 360

Multiply by -1, which changes the direction of all inequalities ( ≤ becomes ≥ ). See * below...

0 ≥ -θ ≥ -360

Add 45

0+45 ≥ -θ+45 ≥ -360+45

Simplify

45 ≥ 45-θ ≥ -315

--

* Maybe you'll see this more clearly if you consider the two inequalities that define the range separately...

0 ≤ θ is equivalent to 0 ≥ -θ
θ ≤ 360 is equivalent to -θ ≥ -360

EDIT: This reversal of inequality happens whenever you multiply (or divide) both sides by any negative number. Be especially careful if you multiply through by a variable IF that variable could be -ve (this might be a something you'll learn about in the future, best to avoid for now)

 

[Vikash]

You start asking about tan(45-θ) = -1, but I think your question is really just about simplifying/ manipulating range definitions. Therefore I'll try to help with the latter, and if you need further help with "the bigger, original, question" then please post back.

0 ≤ θ ≤ 360

Multiply by -1, which changes the direction of all inequalities ( ≤ becomes ≥ ). See * below...

0 ≥ -θ ≥ -360

Add 45

0+45 ≥ -θ+45 ≥ -360+45

Simplify

45 ≥ 45-θ ≥ -315

--

* Maybe you'll see this more clearly if you consider the two inequalities that define the range separately...

0 ≤ θ is equivalent to 0 ≥ -θ
θ ≤ 360 is equivalent to -θ ≥ -360

EDIT: This reversal of inequality happens whenever you multiply (or divide) both sides by any negative number. Be especially careful if you multiply through by a variable IF that variable could be -ve (this might be a something you'll learn about in the future, best to avoid for now)

Dought cleared .....THANKS A LOT??????