trig

[xc630]

Hello I need some help with this trig problem.

In Triangle DEF sec F= -SQRT(2)

I have to find the measure of angle F and tan F


I used tan^2F +1 = sec^2 F and got tan F as 1. But how would you find the measure of angle F?

 

[soroban]

Hello, xc630!

In triangle DEF, \(\displaystyle \sec F\,=\,-\sqrt{2}\)

I have to find the measure of angle \(\displaystyle F\) and \(\displaystyle \tan F\)

If the secant of angle F is negative, it must be obtuse.
\(\displaystyle \;\;\sec\theta\) and \(\displaystyle \cos\theta\) are negative in quadrant 2.

What angle has a secant of -\(\displaystyle \sqrt{2}\) ?

\(\displaystyle \sec\theta\:=\:\frac{hyp}{adj}\:=\:\frac{\sqrt{2}}{-1}\;\) (The hypotenuse is always positive.)

Using Pythagorus, we find that: \(\displaystyle opp\,=\,1\)

Therefore: \(\displaystyle \,\tan F\:=\:\frac{opp}{adj}\:=\:\frac{1}{-1}\:=\:-1\)

The angle looks like this.

       *
       :\   _
       :  \√2
       :    \  θ
      -+- - - *-------
         -1

You're expected to know that the angle is \(\displaystyle 135^o\) or \(\displaystyle \frac{3\pi}{4}\) radians.